11% tongue weight may just be a little too close to the low end and may have crept lower as stuff gets moved around in the trailer, you use up some propane, tank levels change, etc. I'd try increasing your tongue weight up to 13% and see if that works. If this is a rare occurrence, it will be hard to see the benefit right away, but I think it would be a good move.
As for whether you did the right thing, speeding up, or slowing down. I think you got it right. If it is caused by a light tongue weight, it will get worse as you speed up, so while it may help initially to accelerate because it is pulling the "spaghetti noodle" straight, you will end up at a faster speed which could then make the problem worse due to the resultant higher speed. Do as you did, gently slow down by letting off the gas and manually grab the trailer brakes.
LT tires would be a good upgrade as well. If you are within all weights per the scale, then you should be fine with the truck. Of course a 2500 would be nicer, but shouldn't be necessary.
Good luck.