Much heavier people ride various kinds of MTBs that have been reconfigured for riding on the road, and do not experience regular tire problems (blowouts, punctures). You seem to be suggesting that your weight and size is the cause of the tire problems, but that is not consistent with my experience.
As suggested by other respondents, you want some good quality slicks or semi-slicks on this bike to improve rolling resistance. You can run some of them at higher pressures than you might run a knobby tire, which also helps.
Before you install new tires, be sure that your rim tape/rim strip is not the source of your "blowouts." In other words, once rim tape gets tired, and the adhesive doesn't work as well as it used to, the tape can shift around under the tube, exposing the spoke holes, causing flats. Similarly, if you have an incorrectly sized (too narrow) rim strip, you can have the same problem. Bottom line, be sure your rim tape or rim strip is not uncovering the spoke holes.
You did not mention what width of tire you're running on the Marin, but on BikePedia it shows the bike is typically spec'd with something around 26" by 2" so any decent tire around that size should be good.
Schwalbe Marathon Plus can be hard to mount, and they can be "tall" for a given size, but they are about as low-maintenance and puncture resistant as you can get. There are a few different SMP tires available for 26" rims:
Marathon Plus, 50-559, Black-Reflex, Wire
Marathon Plus Tour, 50-559, Reflex, Wired
Marathon Plus, 47-559, Black-Reflex, Wire
Marathon Plus Tour, 47-559, Reflex, Wired
That is what I would suggest, along with some decent-quality tubes.