I figure out songs. That's the only way I practice. I find it makes my ear better.
So, should I write them out in tab? Will that make the song easier to remember or will it just make me to used to tabs?
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Man, you even got the melody right. +1000000 points.

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^thats a brilliant call. *jots in notebook*
I find writing something down makes remembering it easier. It also means you can refer back to the tab, should you not feel like working the song out again.
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I have a notebook somewhere of tabs I figured out. I kept them in case I'd like to look back over them.

Go for it if you have the time.
Even better than tab would probably be writing them on the staff, as in actual musical notes. Bass clef. It'll make you more versatile as a musician--being able to read music and translate it onto the bass is an incredibly important skill (that I'm pretty much just starting to work on).