Too heavy for my tires

Funny you should say that.  I bought some specialized tires and I was getting LOTS of flats - even at pressure....  Weird.

I have some inexpensive conti Ultra Sport training tires that I run at 120psi and they seem great - no issues..

BTW - I'm about 230lbs....

In general, the higher-end race level tires will have softer, more supple rubber and will hold better in corners, especially in the dry weather. The trade-off is tread life. For a bigger guy, it's not at all unusual to square off the rear tire a little faster, or even a lot faster.

One option is to rotate the tires, but it's ill-advised because you really don't want a squared-off tire doing the steering. So the other, more recommended option is to just replace the rear and front as needed. So you may go through 2-3 rear tires before you have to replace the front. 

As for the high-PSI options out there, remember that in order to get any benefit from that PSI, you need to be riding on a very soft, supple tread, i.e. in the 290tpi range. Otherwise the tire will just bounce around and ride slower than the high-psi version. Bottom line is that some deformation of the tire is to be expected while you ride, and as long as you're at the rated PSI for any mainstream tire you should be fine.

 FWIW, I'm 210# and come from a crit racing background--when going around corners at 20-30mph in a pack you get to be a little anal about tires.

What about "normal" tires?

I just started a few weeks back and was budget conscious, just to see if I would like tri'ing, etc .... so bought a $299 hybrid bike at a local 's Sporting Goods store, a Diamondback, Marvista model.

I think the tires say to inflate to 85 psi or something like that.  It has 700 cc tires, not skinny road tires, not knobby tires or true hybrid tires with knobs on the sides and smooth down the middle.  It is smooth (has tread though) and seems to be twice as wide as soom of the high end road bike tires I saw.

Should I keep these tires at or under 85 psi, or can this be pushed up to 10 psi?  I weight about 15 lbs right now and even though I have about 80 psi in them right now, the rear tires esp. seems to be nearly flat when I ride it.  Actually, it is probably flattening out about halfway ....

I gotta ya by 10lbs.  You'll be fine.  I have over 1200 miles on my current set - fyi.