I wouldn't bring any family photos that you don't have copies of, nor would I bring any artwork that can't be replicated....Definitely wash and dry (thoroughly) any items you aren't wearing on a regular basis. We have jeans and sweaters / sweatshirts that we clean once a month - this may be overkill, but since we don't wear them here, we want to be sure they're intact when we need them to go home....
We run our aircon once a day in the evening for dinner, and the AC in our bedroom at night....This has seemed sufficient to prevent mold from growing.
Please don't stress too much about this move - just bring over the stuff your family loves, as well as canned / packaged foods and cleaning products that you thing you may not be able to live without until you acclimate. Most anything can be gotten here for a price, and it sounds like with an $8k - $10k budget, you'll have the $$ to be able to get what you need when you need it.
Welcome to Singapore!!