This can be a Carburetor Problem or an Exhaust Problem.
I Suggest you Check the Exhaust First and then the Carbuertor.
From your Description it Sounds like the Exhaust Port or Muffler is Restricted with Carbon Deposits, the Carburetor Metering Pump Actuator Gap is Incorrect.
>>The Following is a Basic Instruction File for Doing a Diagnostic on the Engine for this Problem. Any Input is Appreciated.
>>**Remove the Muffler and Make Sure the Exhaust Port is Clear.
>>If it is, then Try to Start the engine with the Muffler Off.
>>If it Starts and Runs Properly, then Clean the Spark Arrestor Screen in the Muffler or Replace the Muffler if Required.
Is the Fuel Line from the Fuel Filter Staying Full of Fuel?
The Carburetor Metering Jet Actuator is Set so the Pump Diaphram Dowl is Just Touching the Actuator Arm. These Components are Located Under the Cover with 4 Small Screws Holding it to the Carburetor Body.
Try Setting the Carburetor Air Mixture Screws.
>>The Following is the Instruction File for Setting the Air Mixture Screws. Any Input is Appreciated. The Procedure is the Same for 1 or 2 Mixture Screws, Just Use the Instruction File and Skip Over the Section with the Missing Mixture Screw ( Usually the H Mixture Screw).
>>Now Setting the Air Mixture Screws:
>>****Turn the Idle Air (L) and Main Air (H) Mixture Screws In Until SNUG **Do Not Jam** Then Reverse Both Screws 1 1/2 Turns. Set the Throttle Full Open, Start the Engine.
>>Turn the Main Air (H) Mixture Screw Clockwise Until Proper Revs are Obtained. **Carefully Use your Finger and Move the Governor Arm to Rev the Engine (on Chainsaws and Trimmers Use the Trigger to Rev the Engine) Until you Reach Max RPMs (on Chainsaws and Trimmers, the RPMs are Not Set to MAX. Set to Max and then Reverse the Setting Screw 1/4 or 1/2 Turn. If the H Mixture Screw is Set too High; the Engine will be Starved for the Oil in the Gas and Damage the Piston and Jug/Cylinder) and the Engine is Running Proper at Full Throttle Up a Grade and for Chainsaws and Trimmers they Do Not Bog Down Under a Load. 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 (L) Mixture Screw so there is No Hesitation when Rapidly Throttling from Idle to Full Throttle. Reset the Engine Idle if Required.****
If the Above Did Not COrrect the Problem, then Remove, Disassemble, Soak and Clean the Carburetor.
>>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 have Found that Most People can Use a Breakdown/IPL and Disassemble the Carburetor Enough for Proper Cleaning.
>>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. The Only Parts you May Need to Replace are the Needle Valve and Metering Pump Diaphram.
>>The Metering Pump Diaphram is Under the Carburetor Plate with 4 Small Screws in it.
>>The Dowl on the Diaphram should Just Touch the Metering Jet Actuator Arm.
>>The Actuator Arm is Adjusted by Carefully Bending the Actuator Arm Up or Down to get the Correct Clearance.
>>If you Order a Kit, then Replace All the Old Components with the New Ones in the Kit even if the Old Parts Look Good.
>>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.
>>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.
>>****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 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.