When you say they don't switch off how long do you let them have a chance to switch off, do you have the dreadful air deflector plate between rad and i/cooler they came with.
On a hot day with mine running this set up and perfect it could easily take 5-8 miles once they had come one for fan to go off(I use a "fan" light led).
It will be fine with either 82 or 88 stat, these do get sluggish but would be noticed on a a car as very long time to warm up, the WAHLA stat is the best, twice the cost but worth it.
Whereabouts in SE are you(pm if want), engine DIY on these is childs play, really has got to be one of the most simple engines to do.
Why have you got your relay connected to the SW on the coil, this is a big NO NO, so I would correct that now.
It is entirely possible still to be the relay, some are latchy(sticky), and only need the sligtest amount of current to flow to switch, some switches do leak a slight amount, noit saying this is your case, but I would correct the wiring SW first.
tabetha