From your description, the A/C contactor is unlikely to be bad.

AGC automotive fuses are fast acting.  For electromagnetic loads such as contactors or motors, fast acting fuses are usually the incorrect type and not acceptable.

A time delay or "slow-blow" is probably required.  Look at the VA rating on your transformer and divide by 24.  (i.e. if it is 75VA then the maximum current would be 75/24 = 3.125 A).  Install a fuse sized at or just below the transformer rating that is the slow-blow variety (the type with the heavy or spiral filament).  A low voltage rating is not necessary.  A line voltage rating such as 125v or 250v is perfectly fine.

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  [1]: https://i.sstatic.net/cryFI.jpg