Definitely try increasing the PSI. I'd consider 85-90 on the low end for myself, at 125lbs.
Wider tires would probably help, but I'd take some measurements before going that route. On bikes from that era, there is often not enough frame clearance for much over 25mm (my 2010 cannot go above 23, actually.)
If you ride anyplace where trucks go, sometimes little invisible pieces of wire can get caught. If you turn the tire inside out and rub a cotton ball around, you can find them. I doubt that's the issue, but can be another cause of recurrent flats.