The sound the car makes is as if the sway links were broken, but they aren't. Anyone else has this problem? What is going on? Extra info is that I have air stuck in my cooling system, but my engine never overheats.

 

Have you checked all the ball joints up front? There are more things that can go wrong than just the sway bar links. Struts? Tie rod ends? etc. etc.

How do you know you have air "stuck" in the cooling system? You should have some air in there to allow for room for expansion.