IF your 5th wheel 'DRY' weight is 9600# your tow weight is NOT 10000#...that is only 400#
Typically 1500# + is added weight ---clothes, food, pots/pans, etc etc etc
1800# pin weight......once again that is DRY pin weight based on the DRY trailer weight
You are probably closer to 11,500# loaded up camp ready with a 2400# pin weight
(Really should go to set of scales and get weights)
Those ST235/80R16 tires are capable size tires.....problem is the Brands you have been using are JUNK ==Gladiators, Lionshead, Carlisle
Buy a better quality ST.....Goodyear Endurance or go LT in same size 'E' rated
And WHAT air pressures have you been running in those ST tires that blew?????