I had that "slop" problem with my 2" hitch going into my 2" receiver. The hitch was moving around so much, that it was starting to wear a grove in the shank.
I make this:
There is a 4th side, but I left that off to take the picture.
I have seen where a hose clamp held shims in place around the shank, similar to this:
I would not do any welding to the hitch.....that can mess up the strength of metal.
Buying a new 1 1/2" shank could remedy the problem, but that setup could be "sloppy" also.
Chap , DW Joy, and Fur Baby Sango
2006 Ford F150 Super Cab 4x4
2008 Surveyor 263