Steve, Yes the T-37 is capable of crossing oceans (there have been a few noteworthy circumnavigations) as long as you do a few modificaitons, such as a conversion to a cutter rig, but then again, the route your thinking about, a sloop will be fine. You'll need a larger battery bank than what was factory and some electrical upgrades. You may want to replace the old plastic portlights for these are probably on their last legs. The tankage for a 37 foot boat is respectable, but for a live-a-board, you may want to add more water.
The hull configurations are both stout, but the deep keel version may be more suitable with no centerboard, but the centerboarders will be better for the Bahamas and the Chesapeake. I have the C/B version and feel confident with her sea keeping abilities. All of the ratios, such as the balast/displacement, with the T-37 are in the cruising category and indicate that she is stiff and can handle the loads associated with cruising.
For the comparitive money, the T-37 plus the upgrades makes for a good buy with the added benefit of having a beautiful classic that can sail !!!
Good luck, Tom