Unfortunately the only way to tell if it's already damaged is to test it in another system, if you know someone who is happy to let you try it in their system, go for that. As to the fan, generally the fan tends to go first, it is very unusual for only the
cpu fan power section to go. You could try the fan connected to the case fan power point, but, take the
cpu out before you do that as it's a sure way to blow it by running it without a fan.
What processor do you have so I can make some better recommendations as to what sort of fans you should be looking at?