With my sink originally, when the sprayer was in-use, water would stop flowing through the main faucet and the sprayer would spray at pretty high pressure.

Now, the sprayer will spray at a low-pressure, and the main faucet will stay on (and also operate at a low pressure). Not using the sprayer will result in the main faucet having high pressure (as intended). The same results occur whether hot, cold, or both kinds of water are used (lack of better wording).

If I recall right, this started happening after somebody basically kept pressing and releasing the sprayer button repeatedly. Anyone have any idea if something like this might be easily fixable?

Inside the barrel (the part that swivels) of the spout is a little valve that stops the water from coming out the spout when water comes out the sprayer. It is called a diverter valve.
Your diverter valve may be defective or it may be clogged, preventing it from closing the spout.
To repair you must remove the spout and clear or replace the diverter valve.

Agree with Hkstroud... see images below.