It's also possible your yeast pitching rate is off and you're getting a slight bit of acetaldehyde (green apple flavor). At a low level that could be giving you the puckering quality.
As far as the fusel angle, I'm not sure what strain of yeast you're using but even a 10 degree temp increase would only put you at 72*F. That temp is a bit high but it shouldn't be giving you a huge fusel bomb, and you would notice the flavor mellowing a bit over time if that's what it was.
Not saying it couldn't be but i would suggest getting yourself a cheap stick on thermometer to get a better read on the actual beer temp and maybe looking into the yeast health and pitching rate.
The fridge or freezer would be a nice upgrade. A hassle free, temp controlled fermentation chamber is really convenient and takes alot of guess work out of things, but it wouldn't necessarily solve this problem.
My 2 cents.