Transformer to central air is burning out but 3 amp fuse is still good. Any ideas?

That's an odd one. Is this A/C only, A/C on a furnace, or a heat pump?

Actually a 3 amp fuse is almost a 75 VA transformers capacity
3 amps times 24 volts = 72 VA
If you have a smaller transformer than a 75 VA this explains it to me.
For continuous a 75 VA transformer is - 20% od its rated VA
So that 3 amps is really 2.4 amps continuous, take an amp draw and buy a 75 VA transformer with a circuit breaker built into it they are about $50.