From your Description it Sounds like a Carburetor Problem.
>>The Following is a Basic Instruction File I made for Checking the Carburetor Solenoid. Any Input is Appreciated.
>>**Check for Current/Voltage (10.5 VDC Minimum) to the Carburetor Shut Off Solenoid. If the Carburetor Shut Off Solenoid has Current, then On Some Solenoids you can Remove the Needle and Replace the Solenoid to Test them.
>>****PLEASE DO NOT OPERATE THE MOWER WITH THE SOLENOID BYPASSED IN THIS MANNER. THIS IS "ONLY" FOR TESTING.****
>>For Others, Remove the Carburetor Solenoid and Go to the Hardware Store. Purchase a Bolt of the Same Diameter and Threads. It Needs to be 1/2" to 3/4" Long for the Horizontal Mount Solenoids and 1/4" to 1/2" Long for the Vertical Mount Solenoids. Put this Bolt in Place of the Solenoid. If this Corrects the Engine Problem, then Replace the Solenoid.
>>***NOTE: THIS IS FOR CHECKING THE SOLENOID ONLY I NEVER SUGGEST YOU OPERATE THE ENGINE WITH THE SOLENOID OUT OF THE CARBURETOR EXCEPT FOR TESTING PURPOSES***
>>Sometimes the Plunger is Moving, Just Not Far Enough to Allow the Gas to Enter the Jet Properly.** **Use 440 Wet Dry Sand Paper and Engine Oil to Clean the Pin. Wet the Sand Paper with Oil Until it is Flexible and then Lightly Polish the Pin. There is Some Slop/Free Play in the Solenoid Pin. Usually the Electromagnet is Simply Not Pulling the Pin as it Should or there is a Small Bur on the Pin. Use 440 Wet Dry Sand Paper and Engine Oil to Clean the Pin. Wet the Sand Paper with Oil Until it is Flexible and then Lightly Polish the Pin. Clean and Lightly Oil the Pin before Inserting it Back into the Solenoid. Just a Film of Oil is All it Takes.**
If the Carburetor Solenoid is Good, then
>>The Following is a Basic File I Made for Cleaning Carburetors. Any Input is Appreciated. Even though the Carburetor Looks Clean, the Internal Passages May be Restricted with Varnish that Gas Causes to Build Up over Time.
>>Spray Cleaners Remove this Varnish in Layers, so Soaking is the Only Sure Way to Remove ALL this Varnish.
>>I Suggest you Make Sure the Gas is Free Flowing to the Carburetor Inlet Port (if the Model does Not have a Fuel Pump). If the Gas is Free Flowing to the Inlet Port, then
>>**Remove, Disassemble and Soak the Carburetor Overnight in Carburetor Cleaner (I Use Gunk).
>>The Primer Bulb is Held in Place with a Locking Collar at the Base of the Primer Bulb. Use a Small Flat Screw Driver and Carefully Pry the Lock Ring Up in Small Amounts All Around the Lock Ring until the Lock Ring and Primer Bulb can be Removed from the Carburetor Body. Usually a New Primer Bulb is Required when you have to Soak the Carburetor. The Rubber is Usually Rotted to Much for the Primer to be Reused.
>>Your Primer Bulb May Differ Slightly, but Removal is Usually Almost the Same Except for the Primer Bulbs that are Bolted to the Carburetor. These are Simply Unbolted and Changed with the new Primer Bulb.
>>I have Found that Most People can Use a Breakdown/IPL and Disassemble the Carburetor Enough for Proper Cleaning.
>>You Only have to Remove the Bowl (if Applicable), Float Pin (if Applicable), Float (if Applicable), Needle Valve and Needle Valve Seat if Applicable. This is a Flat O-Ring Located in the Inlet Port the Float Needle Rests in. Use a Strong Wire with a Slight Bend to Remove the Old Seat. Sometimes a Small Pocket Screw Driver can be Use Also.
>>The New Seat will have a Slightly Beveled Side and a Flat Side. The Flat Side goes Towards the Carburetor Body when Inserted into the Inlet Port. Spray the Seat with Spray Lubricant to make Installation Easier. The End of a Drill Bit that is Slightly Small than the Seat can be Use as an Insertion Tool.
>>Wear Gloves or Use a Rag when Handling the Sharpened End of the Drill Bit. Remove Any Adjustment Screws that Go Into the Carburetor Body. The Welch Plugs Do Not have to be Removed.
>>If you are Able to Remove and Disassemble the Carburetor and Keep the Gaskets Intact, then Usually these Gaskets can be Reused. Even the O-Ring Around the Bowl can be Reused if it is Not Broken. The Only Parts you May Need to Replace is the Float Needle Valve (and Seat if Applicable).
>>The Float Needle Usually is Not Replaced Unless it is Not Operating Properly (the gas flow not shutting off and the carburetor is Flooding). Do Not Remove the Main Nozzle. This is a Pressed Fit and Removal is Not Required for Cleaning. Soaking and Blowing the Carburetor Out After Soaking will Clean the Nozzle.
>>Once Disassembled, then Soak the Carburetor Body and Parts in a 1 Gallon Can of Gunk Carburetor Cleaner Overnight. The Can has a Parts Tray Inside it for the Small Parts.
>>Once the Carburetor and Parts are in the Cleaner, you can Replace the Lid for Safety and to Prevent Accidental Spillage. Then Blow Out the Passages with Compressed Air.
>>***(Do Not Use High Pressure Air for the Zama Carburetors, they have Check Valves for the Primer and these are Usually Blown Out of the Carburetor if Not Careful. Allow the Zama Carburetor to Set on a Drip Pan and Dry)***.
>>****All the Carburetor Adjustments are the Same for Lawn Mower and Trimmers. There are Several Location for the Air Mixture Screws. The Idle Air Mixture Screws are Usually Located at the Top of the Carburetor Bowl and the Top of the Carburetor Body.****
>>If the Main Air and Idle Air Mixture Screws are Side by Side on the Side of the Carburetor, then the Idle Air is Nearest the Engine. The Main Jet Air Mixture Screw is Located in the Bottom of the Carburetor Bowl or Beside the Idle Air Screw on the Side of the Carburetor Body.
>>Some of the Older Model Carburetors have the Main Jet Mixture Screw Straight in from the Top of the Carburetor Body (this is Rare anymore). If you have Cleaned the Carburetor (Disassembled and Soaked Overnight in Carburetor Cleaner (I Use Gunk). Then Blow Out the Passages with Compressed Air and Install a New Kit if Required. Now Setting the Air Mixture Screws:
>>****Turn the Idle Air and Main Air Mixture Screws In Until SNUG **Do Not Jam** Then Reverse Both Screws 1 1/2 Turns. Holding the Throttle Full Open, Start the Engine. Turn the Main Air Mixture Screw Clockwise Until Proper Revs are Obtained. Occasionally you May have to Turn this Counter Clockwise to Achieve the Proper Revs. Now Allow the Engine to Idle. Set the Engine Idle Screw (Not Idle Air) so the Engine will Stay Running if Required. Now Set the Idle Air Mixture Screw so there is No Hesitation when Throttling from Idle to Full Throttle. Reset the Engine Idle if Required.****
>>If This Carburetor has a Single Air Adjustment (Except Tank Mounted 9200 and 100900 Engine Model Carburetors), Use the Section Above that Pertains to Full Throttle RPM Air Mixture Screw and then Adjust the Screw if Required to Eliminate Any Hesitation when Throttling from Idle to Full Throttle.
>>For the 9200 Model Tank Mounted Carburetor Adjust the Air Mixture Screw Full In, then Reverse 1 1/2 Turns. Set the Throttle Lever to Full Throttle and Start the Engine.
>>Now Carefully Use 1 Finger and Open the Throttle Plate and Over Rev the Engine Slightly. If the Engine Over Revs and Does Not Struggle to Over Rev, then the Carburetor is Set. If the Engine Struggles to Over Rev, then Turn the Adjustment Screw In 1/4 Turn and Repeat the Over Rev Test.
>>If you Adjust to 1/2 Turns In and the Engine Still Struggles to Over Rev, then Return the Adjustment Screw to 1 1/2 Turns Out from Snug and Turn the Screw Out 1/4 Turn. Do the Over Rev Test. Continue this Process Until you have the Engine Over Revving without Struggle.
>>By Adjusting the Carburetor on this Style Carburetor Until the Engine Over Revs without Struggle, you have Adjusted the Air Mixture to the Best Possible Setting.
>>This file was Intended to Give you the Basic Carburetor Cleaning Instructions and May Not Reflect Your Carburetor Components.
>>If you have Questions, Please Ask. The links above Provide Good Directions on Cleaning the Carburetor. Make sure you use an Compressed Air to Blow through all the Carburetor Passages to make sure they are Clear.
Please, Do Not Hesitate, If you Hit a Snag or this Does Not Correct the Problem, I am Here if You Require More Assistance.
Hope this Helps. Let me Know What Happens, Please. May the All Mighty Bless You and Yours. Be Safe and Be Happy. Thanks.