I agree that they should. So I need to buy new batteries. What to buy?
nothing I've tried has worked well.

I'd try another set of energizer NiMH AA's. I have 2 sets for my
digicam (3-yr old Canon PowerShot, takes 4xAA batts), and it easily
gets 200-300 shots per battery charge. I just use the charger that
came with the batts.
There are now several brands that advertise 2500 mAh AA batteries
(Energizer, Sony, and other well-known names), and I think ours are
2000 or less, so the new ones should theoretically be even better.
As far as I know, basically all the decent chargers now are "smart" so
you should be able to leave the batts on without a problem.
You might also try contacting the camera manufacturer - my uncle had an
older Kodak camera that went through batteries after just a couple
shots, got it replaced twice, and finally got one that had reasonable
battery life.
Good luck,
Andy