It's a Kenmore Model 106-56712500 , can't shut off alarm after changing filter, and now won't make ice or dispense water! How do I do a proper re-set code?

There should be a shut off valve inside where the filter goes. When you remove the filter, it should slam shut.

https://www.google.com/search?q=Kenmore+Coldspot+gushes+water+when+changing+the+filter

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Assuming you followed the water line installation directions and you have water available to your new refrigerator/freezer here is the rest of the "priming" info you requested. (basically it tells you how to run a bunch of water through the system by filling cups or glasses to remove any debris and/or air...

Automatic Ice & Water Dispenser
Priming the Water Supply System
1. Begin filling the tank by pressing and holding a drinking glass against the water dispenser paddle. 2. Keep the glass in this position until water comes out of the dispenser. It may take about 1½ minutes. 3. Continue dispensing water for about three minutes to flush the system and plumbing connections of any impurities (stopping to empty the glass as necessary).
Ice Maker Operation & Care After the refrigerator is installed properly and has cooled for several hours, the ice maker can produce ice within 24 hours. The ice maker will fill the ice bin from the rear. You must dispense some ice to force the ice forward in the bin. This will allow the bin to fill completely.
Turning the Fresh Food Ice Maker On and Off Ice production is controlled by the ice maker's On/Off icon on the control panel. Press and hold the "Ice Off" icon for three seconds to turn the ice maker "ON" or "OFF". The ice maker is turned off when the LED above the "Ice Off" icon is illuminated.
Using the Ice Maker after Installation Before making ice for the first time, be sure to prime the water supply system. Air in new plumbing lines can result in two or three empty ice maker cycles. Furthermore, if the system is not flushed, the first ice cubes may be discolored or have an odd flavor.
CAUTION
For proper dispenser operation, recommended water supply pressure should fall between 30 psi and 100 psi. Excessive pressure may cause water filter to malfunction.
NOTE The water dispenser has a built-in device that shuts off the water flow after three minutes of continuous use. To reset this shutoff device, simply release the dispenser paddle.
IMPORTANT On occasion, unusually small ice cubes may be noticed in the bucket or in dispensed ice. This could occur in normal operation of the ice maker. If you start seeing these more frequently, it may be an indication of low water pressure or the water filter needs to be replaced. As the water filter nears the end of its useful life and becomes clogged with particles, less water is delivered to the ice maker during each cycle. Remember, if it's been six months or longer since you last changed your water filter, replace the water filter with a new one. Poor quality household water may require the filter to be changed more frequently.
IMPORTANT Your ice maker is turned on at the factory so it can work as soon as you install your refrigerator. If you cannot connect a water supply, set the ice maker's On/Off icon to Off and turn off the water supply valve; otherwise, the ice maker's fill valve may make a loud chattering noise when it attempts to operate without water.
Hope this helps. Randy

Hello nott225 - The issue sounds like it might be related to the temperature in the freezer. What is it set to? Make sure that the temperature is approximately around 8 degrees Fahrenheit or lower. If the freezer is too warm, the ice maker won't cycle properly. If it is properly set, then consult a professional to accurately pinpoint any potentially more serious issues like a faulty ice maker valve or even the ice maker itself.

what is your model number is then we can get started then

It is definitely the water filter. With the hardness of the water from our county I have to change our filter every 6 months. After replacing the filter be sure that it is seated well. It can leak if it is not fully engaged. Run water through the dispenser until the air is out.

We also have a kenmore side by side refrig and had the same problem,,,
FIRST, CHECK THE CONNECTION BETWEEN NEW FILTER AND ATTACHMENT POINT,,
IF THE RUBBER GASKET CAME LOSE DURING THE CHANGE, IT MAY BLOCK THE PATHWAY FOR WATER TO FLOW
NEXT, CHECK THE WATER/ICECUBES/CRUSH LEVER SWITCH TO MAKE SURE IT IS NOT PART WAY BETWEEN ONE OF THE SETTINGS,
NEXT,CHECK TO SEE IF YOUR UNIT HAS A RESET BUTTON,
THE RESET BUTTON IS USUALLY IMMEDIATELY INSIDE THE OPENING WHERE THE WATER/ICE DISPENSES AND UP, YOU WOULD NEED TO PLACE YOUR FINGER RIGHT DIRECTLY UP BEHIND THE FRONT PLASTIC COVER, AND IT'S GOING TO BE A VERY SMALL BUTTON, I AM NOT SURE WHICH SIDE ON YOUR UNIT R OR L, BUT DIRECTLY UP IN THE CORNER,,
I HOPE ONE OF THESE HELPS !! angi

It is possible the water control valve for that dispenser has failed and will not open. This happens a lot for those and the ice maker valves. All you need is the model number for your refrigerator and that will look like this for Kenmore. 213.564897 for example. and I will help you find the correct part number. Sea Breeze

your freezer had a heater and a bi metal behind its pannel. if they won't work properly than you would face problem like this.......contact your service authority. it cannot be solved by your own

the indicator light could be on the control panel inside the refr section if your controls are there, or it is most likely on the front of the dispenser panel. it only comes on when you draw water from the dispenser and thats when you will see it be green, filter ok, yellow, filter at 80% life, or red, time to replace the water filter.
also you dont need to shut off the water when changing the filter as it is made with a shut off device in the filter housing already and is automatic.
by the way filters are expensive, go online and check prices. google "water filters" and find a good price.

there is valve for each right at the water inlet, your valve is stuck or your coil is bad......get a new valve.