A better (or at least properly working) brake controller would, in my opinion, be an essential. There are reasonably decent ones available for not excessive amounts of money. You should certainly be able to apply them without locking them up; a decent (or even not so decent) brake controller will have some sort of a gain or sensitivity control to adjust this. You want a proportional brake controller, not a time delay one, if at all possible; there are several ones available for less than $100 online. (I don't know enough to recommend any specific models.)
Your handling issues do sound, at least in part, like they might possibly be due to insufficient tongue weight on the trailer. The only way to know for sure is to do some weighings to determine the trailer weight and the tongue weight and your truck axle weights. Those numbers, and the axle and combined weight ratings, would help inform what sort of changes would be most sensible to make and what tire pressures you should be using and so forth. Without them, any attempts at diagnoses are pretty much just stabs in the dark. (They'd also help tell if you do have too much trailer for the truck or not.)