Why are the batteries discharging so quickly?

Sounds like you have a capacitor going bad or it dried out. Though I'm not familiar with this camera so maybe get a teardown and look for bulging capacitors.

As I remember, these use tantalum caps, so they are unlikely to be bad.

I used to repair these, but it has been many years, so I don't remember what the current drain problem (rare) was. I do know that working on SLRs requires much more than the usual "handiness" that many people have. Unwise use of solvents, brute force in opening, or even just touching some things can cause expensive damage.

You can get a service manual online (e.g., ++https://www.manualslib.com/download/7040..... If you are up for it, give it a try. You need electronics troubleshooting skills and equipment as well., Go slow, patience is your friend. If you take it to a camera repair shop, make sure it's a good one. Those are thin on the ground, it really depends on where you live.

You likely have a capicitor that needs replacing, or perhaps the motor drive is running (continually focusing) even if you aren't using the camera. If you put the battery in, and leave the camera off for several hours/days does the battery drain at all? I'd suggest taking your camera to a camera store or you could call Canon to see if they can repair it for your (as a mail in repair)