Could a bad thermostat be causing my air conditioner to pull too much voltage?

It is receiving a low voltage signal from somewhere else. Its powered by 24volts. And its receiving something less than 24volts (like 16-20v) but its enough to build up enough to want to pull in the contactor. I've seen it before but can't remember what the cause was. Could be miswired somewhere or a fault at the t-stat. If it has a circuit board they can get a gecko or bug behind it and the metal cabinet that can cause a "short" that will allow voltage to flow through and power the circuit that feeds the contactor.

Go to the air handler and look near the transformer for a 3 amp blown fuse. Pull it out and put a new one in (it resembles a fuse used in cars).

The only way your thermostat cannot be energized is if the red wire to the thermostat is broke between the thermostat and the transformer, the tranformer is bad, there is no power to the air handler, or the low voltage fuse is blown.

It sounds like your contactor is not pulling in. This can be caused by multiple things such as a clogged drain on the inside Air Handler Unit tripping the float switch which would cause the compressor to stay off, a bad thermostat, improper refrigerant charge (low pressure or high pressure switch located in the outside condensing unit), tripped breaker on your line voltage side feeding your condensing unit, bad fuses in disconnect, or possibly a bad contactor (burnt points, bugs in the points, or shorted magnetic coil.)

Check the fan relay. These relays are used to cut the fan on and off when the thermostat is set in the air conditioning or heating mode or when you move the thermostat to the "Fan On" position. The low voltage usually the green thermostat wire is wired to terminal #1. The low voltage common wire to terminal #3. One line voltage wire is attached to terminal #4. The high speed blower wire usually black is attached to terminal #2. When the thermostat calls for air conditioning or "Fan On" the 24 volt coil is energized, the terminals 4 and 2 close. This causes the fan to come on in high speed. When your home cools down and the thermostat cuts the air conditioner off, the low voltage coil terminals 1 & 3 are denergized, and the fan goes off. Sometimes these little relays stick which causes the fan to run all the time or will not make good contact, which causes the fan to not run in the "Fan On" or air conditioning mode.

Check your disconnect or breaker that is located close to the outside unit. Next you need to check the contactor in the outside unit. Set the thermostat to make the air come on then check the contactor. It should be pulled in closing the contacts. If it isn't the problem could be the contactor or thermostat. Next check the voltage on the load side with it pulled in. If that's good then check the capacitor. If it looks swollen, it's bad. You can also use a multimeter with the capacitor check function to do this. Let me know what you find.

First check the circuit breaker or fuses. Remove the cover on the outside unit that covers the contactor, circuit board, and capacitor. Check for voltage on the line side of the contactor. Set the thermostat to make the air come on then check the contactor to see if it is pulled in (contacts closed). If it isn't pulled in then check for 24 vac on the contactor coil. If the contactor is pulled in check for voltage on the load side of the contactor. If you have it there then check the capacitor(s). Some units have 1 that runs the fan and compressor and some units have 2 ( 1 for the fan and 1 for the compressor). If you don't have 24vac at the contactor you can install a jumper at the thermostat between R and G if it is just an air conditioner or R and Y if it is a heat pump and see if the thermostat is bad.

By relay, I am assuming you are talking about the contactor. First, you will need to visually inspect it for scorching or pitting. This is caused by wear and tear and can cause the unit to draw more amps or not make proper contact and only allow a certain percentage of the voltage through it. To check operation, the coil should pull in when 24V is supplied to it. If 24V is being supplied and its not pulling in, you can attempt to manually depress it with a non-conductive item, such as a plastic pen. If the contactor stays pulled in, then you either have less than 24V indicating a mis-wired or failing transformer or you have a weak contactor coil. In which case, the contactor needs to be replaced.

If no voltage is present at coil, then your problems lie elsewhere. Maybe within the thermostat, a float switch, or bad wiring.

The fan should come on when the compressor comes on. If it doesn't, you can attempt to spin it by hand really fast. If the motor begins to run at this point, check the capacitor. Be very careful if attempting this. The motor spins at close to or in excess of 1000 RPMs and if it starts up, you don't want your hand close to it.
If the motor fails to run, again check the capacitor and supply voltage. If both previous check pan out, then the motor is bad and will need to be replaced.

You should have 24v ac at the coil to the contactor (points) as you call them. The 24v comes from the indoor unit through the thermostat. If you manually push in the contactor does the unit run. If so, then the problem lies in the low voltage circuit somewhere.
If the indoor blower comes on when you switch the thermostat to fan them you have low vaoltage and the transformer is okay if not then change the transformer or fuse. if you have low voltage then you need to be checking for a broken wire of a bad thermostat on the cooling side. depending on your thermostat a relay can go bad in it the runs the cooling.