Air Asia is very strict with carry-on items. Yes, they will weigh your carry ons.... On top of that, I doubt that you can interline bag transfers between Singapore Airlines and Air Asia. They don't operate from the same terminal, too. So if anything, you'll need to clear immigration and customs at SIN and re do the entire check in again for the Air Asia flight.

Air Asia has no transfer desk for transit passengers.... Unless you're going between their flights like in KUL. Your separate tickets won't help your situation, too.