I agree that it's likely to be the battery's fault. You said it was dead and you recharged it. Not so fast. Once a battery goes dead...really dead...it often cannot be brought back to life.
A poor battery, as CLR said, will suffer a significant voltage drop when you add the load, and as voltage drops, amps increase. High amps blow fuses.
While you're at it, make sure every connection from the battery to the winch is clean and tight. Poor connections cause issues, too.
So, it's pretty clear your battery is shot. Start with a new one.
Your old one can be turned in for the core charge.
If you have a
good battery charger that plugs into 120 volts, you can "jump start" your winch when connected to the new battery and risk your $10 fuse to make sure that good voltage and current do the job. You could accomplish the same thing with jumper cables from your running car. LOTS OF 12 VOLT POWER RESERVE.
If that works, problem solved. If that DOESN'T work and you continue to blow fuses, there's something wrong with the winch. But, important note. These winches can be driven manually with a hand crank or a cordless drill...in other words, it shouldn't take 60 amps to make this thing go. If it continues to be sluggish, then fix the winch.
Now the last. DO NOT LEAVE YOUR BATTERY CONNECTED when storing your rig. Make sure you know your wiring connections to the battery, and disconnect the battery completely when storing it. Even better, bring it inside and put it on a
maintainer charger. Every rig has parasitic loads that can drain the battery in between 10 to 20 days if not on a charger. Disconnected, a fully charged battery can do just fine all winter.