I'm looking for some thoughts on a situation with my cat...

I have an 11 year old indoor, neutered (overweight, but not obese) male
cat. He went to the vet recently for a yearly check-up. I mentioned that
he's seemed lethargic lately and that it seems like he's drinking/urinating
more than usual, and also that I'd like to get some blood tests done. So,
the vet  did some tests (in the clinic) for liver enzymes and blood glucose
and one other I can't remember, related to kidney function I believe. The
results of tests regarding his liver were quite high and the others were
normal. He started on Hill's l/d food then. I was told to return in a
couple months and get him checked again.

Less than a week later, I took him to a different vet for a second opinion.
The second vet noticed redness and swelling in the back of his throat on
both sides and we started him on some antibiotics. The second vet ordered a
full "geriatric blood panel" from a laboratory. The results of which were
well with in the normal range for liver and all the other tests (including
thyroid T4), excepting some high cholesterol and a very slightly elevated
red (yes, red not white) blood cell count. She didn't think it was possible
that his liver repaired itself in a matter of days, but more likely that
the other blood test was incorrect.

The antibiotics seemed to have helped, and he appears to be perking up.
But, I'm still worried something could be wrong with him. He is going back
to the second vet this week (along with my other cat, who appears healthy,
but I want him to have a second check-up with the new vet) and I'm thinking
that I'll ask her to run (at least) the liver tests again just to be sure
his liver is really ok. But, I'm still very concerned and wondering if
there's more I could do to make sure he's alright...? Anything I should be
sure to enquire about when we return to the vet...?

I want to get him losing some weight, but if he has liver problems that
doesn't seem a good idea. I also don't want to cause him undue stress by
having extraneous tests. I'm just terribly worried and slightly confused.

Thanks for any help!
=^.^=