The water jug cleaning of your Thompson Center Hawkins should work just fine. Perhaps a few solvent patches to make sure you got it all out, and are nice a clean of all fowling. Then I'd run some isopropyl alcohol patches down the bore to displace any moisture in there, finally some dry patches to me positive that the bore is nice and clean. After that some quality gun oil like Birchwood Casey Sheath should protect the rifle.

Go ahead and sight it in. It should be real close to your POI when you go hunting. I shoot mine all summer in the heat and humidity and have not seen too much of a difference when hunting. Like you said, at least you're close to your hunting so that is a good thing. Study hard and hunt hard and it will all work out. Time passes fast when you keep busy... Good luck this hunting season.