1) It sounds like you are only cleaning with Hoppes #9. This is insufficient. First, get something rated for carbon and get all of that out alternating a brass brush (from the breech! preferable with a bore guide) with a couple patches. Once the carbon is out, put a couple patches on your jag (taken off the rod) and use it to plug the muzzle; careful with your crown. Stand the gun up on something muzzle-down. Using a straw, fill the barrel right up to the chamber with a good copper remover like the Barnes CR and let it stand an hour. Clean again with a couple Hoppes patches to get ALL the copper solvent out and oil. I have a sub-.5MOA 300WM that will shoot 4in groups if I fire long enough without cleaning it this way.
2) If you shoot 3 shots then wait each shot experiences a higher temp than the last and you'll have a long wait at the end to get it back down to temp. Instead, wait 3-5min between shots (depending on air temp) each time. Each shot will experience roughly the same conditions in the chamber and improve your consistency.
3) Float the barrel. This alone could solve some of your problems with the heating by removing stock contact. Easily done with sandpaper.
4) I recommend doing pillar and glass bedding at the same time. If you have a torque wrench for your action screws, you can elect not to install pillars and you should be "ok".
5) You are handloading? If this load is shooting so terrible it sounds like it was not worked up for this gun. Work up a load for this gun specifically. Figure out your OAL to the lands, back off a few hundredths, and work your way up with the powder until you see pressure signs then back down half a grain or so. Load 3 rounds at several different powder charges across a 2-3 grain range (under max) and see where the sweet spot is.