I have put more air in than the max PSI on a tire before, I have never had a tire blow out. Experiment with sneaking the pressure up. Start at 100 pounds, then if you feel an improvement after a few days go up to 110 pounds.
At a bike + body weight of 340 pounds, you don't want 25mm or narrower tires. Are these your tires?
If so, you can certainly find something that is more of a slick than these. You might also go a bit wider than 30mm, say 35's if your bike with take them. That way you can run a bit lower tire pressure but still have plenty of air to be firm. I have had good luck with these:
http://www.performancebike.com/bikes...86513__1586513
Certainly not flat proof, but the kevlar bead goes a long way toward slowing down road debris. I bet others could recommend more flat proof tires which are slicks.
I congratulate you on getting out and riding to loose weight. The best way to improve speed will be to just stick with it, and don't fret over tire selection all that much. You can loose a lot more bodyweight than you can tire weight.
Summary: Try increasing the pressure in the tires you have now, when you need new ones, or just want to try new ones, go for a bit wider and slicker, and keep riding as you will loose bodyweight which will make you faster. I am rooting for you!!!