My 2 fancy goldfish have broken tails, do they just heal over time?

That sounds like fin rot. It's caused by bad water quality. You should test your water and do more water changes. Also if you tank isn't big enough or your filters aren't strong enough it could be causing it. For 2 fancy goldfish you need 40 gallons with 80 gallon filtration. Do not feed them more, if anything feed less. Fish don't get ich and fin rot for no reason, so you need to find out what's wrong with your tank.