I do not have the songs that are on my iPod on my computer because I don't want to clutter up my hard drive on my computer
I want to echo what a really, really bad idea that is. Corrupt iPods are a fairly common problem and a restore is required. That means everything on your iPod will be erased. What will you use as a back up to reload all those songs any other important data?
Anyway, here are some Windows programs that will transfer songs from your old iPod to your iTunes library and from there you can transfer them to your new iPod.
For iTunes version 7 or later, then you can transfer purchased iTunes store music from the iPod to an authorized computer by using the "file/transfer purchases from iPod" menu. Note that the maximum of 5 authorized computers applies here.
Find out how to do that here.
How to copy iTunes purchases from an iPod or iPhone to a computer.
For all other non purchased content (your own CDs etc), check out the instructions/suggestions here.
Music from iPod to computer (using option 2). This a manual method using "hidden folders" and although it works, it is a little more involved than other methods.
Much easier ways are to use one of the many 3rd party programs that copy music from the iPod to the computer.
One of the most recommended is
Yamipod. This is a free program that transfers music from iPod back to the computer. However, it does not transfer playcounts/ratings etc.
Other free programs are
Pod Player,
SharePod and
Floola.