I have found over the years that a more road performance tire is better. I used to tow with a 1996 Buick Roadmaster station wagon. It had aftermarket rims with 245/60R17 all season Michelin's on it. They were okay, but then I switched to 235/65VR17 tires that were Bridgestone high performance summer tires. They were a huge improvment. With my larger trailer and 3/4 ton Suburban, I originally had 245/75R16 Michelin LTX2 All season tires, they worked really well. I then bought aftermarket rims and put on Cooper AT3 tires in size 265/65R18. Both were load range E. The All terrain tires have a LOT more tread squirm and are not as stable. But they are much better in sand!
As for shocks or struts, the firmer the better. But most importantly, make sure they are in good condition. If they are original, I'd consider replacing them if you plan to keep the vehicle, even if they seem okay. They have probably degraded. If they really seem fine, just try them first and see how they work.
2014 Wildwood 26TBSS - Upgraded with 5200lb axles and larger Goodyear ST tires
2003 Chevrolet 2500 4x4 Suburban 8.1L 4.10's