I ventured up into the mountains a few weekends ago and couldn't pass up the only mud puddle I came across. In the deepest part, it came up to the top of my 31 inch tires. I went through it with no problems, but covered my engine with mud in doing so. My alternator lights came on about 10 minutes later, so I parked it after I got home and let it cool down, and sure enough it wouldn't start back up. I replaced the alternator the next day, but noticed it wouldn't start up right away. I would turn the key and hear a click, then nothing. However, if I kept trying to start it, it would fire up eventually and run fine.

After a week of inconsistent starting, my alternator lights came on again and I noticed the bolt on the top connection to the alternator had melted to the terminal.

Before I replace the alternator again, should I be concerned about a short somewhere? What's the easiest way to track it down if so?

Yeah, you should be concerned about a melted alternator terminal.
Get yourself an ohmmeter and a wiring diagram and start at the melted wire.