Have you tried reducing the psi? My analog mtb has 2.3 tires that are set up tubeless on i35 rims and I run 15psi in them all the time. They do really well in the desert environment, sand especially. My e bike has narrower 1.75 tires on i25 rims set up tubeless and I run 35psi in them and also have had good luck in desert terrain.
Going tubeless really helps to be able to run lower psi as they take away the risk of pinch flatting plus self seal when necessary. At the very least trying to find the lowest psi you can run without flatting, even with tubes, will result in a better ride than I feel you are experiencing currently it you think a wider tire will be any help.
Take your caliper and measure the chain stay and fork width. Be certain to check the brakes. When I went from 1.95 to 2.5 I found I have to completely release the V brakes. I have about 3/8" clearance on both sides of tire front and rear.