Why does my resistance readings keep decreasing when I attach my multimeter to the circuit?

your multimeter is bad because the resistance should be constant.

Does your multimeter ever level out? If it does, then that is your reading. If not, go to radioshack and buy a resistor of any kind. The idea is you know how much resistance the resister has, and if your multimeter reads correctly, then you know your multimeter works correctly. Electrical is funny sometimes.
Another thing is, do both terminals lead back to each other? What I mean is this:

Terminal 1-------resistance----------------terminal 2

OR

Terminal 1-------------------------------resistance-------------------ground
Terminal 2-------------------------------resistance--------------------ground

Resistance measuring will only work if you use the first drawing, from the way you are testing. Now if you poked terminal one and poked in between resistance and ground, you'd be able to measure correctly as well. Good luck, email me with any more questions!

You do not use the resistance setting for checking the circuit. Use either the " DC Amp" connection or the voltage setting.

Use the resistance setting for testing fuses out of circuit.

The meter is bleeding off any stored current in the circuit.It is providing a circuit path to ground from any capacitor.