*How can I get my new music onto my iPod without erasing the music that is already existing on it?*
If you simply want to continue using the iPod without syncing (and losing the contents) then change it to manual management. When you connect to your new computer and you get the message that it is linked to a different library and asking if you want to link to this one and replace all your songs etc, press "Cancel". Pressing "Erase and Sync" will irretrievably remove all the songs from your iPod.
Alternatively open iTunes first and go to Edit>Preferences>Devices and check the box "Disable automatic syncing for iPhones and iPods" and choose Ok. Then connect your iPod.
When your iPod appears in the iTunes source list check the "manually manage music and videos" box in Summary then press the Apply button. In manual mode you can drag and drop new content to the iPod from iTunes without erasing what's currently on it:
Managing content manually on iPod and iPhone
If you want to recover the music from the iPod and transfer it to your new computer there are a few things you can do. iTunes will only give you the option to copy your iTunes Store purchases directly from an iPod to the computer, you'll find details in this article:
Copying iTunes Store purchases from your iPod or iPhone to a computer
For everything else (music from CDs, other downloads etc) there are a number of third party utilities that you can use to retrieve the music files and playlists from your iPod. You'll find that they have varying degrees of functionality and some will transfer movies, videos, photos, podcasts and games as well. You can read reviews and comparisons of some of them here:
Wired News - Rescue Your Stranded Tunes
Comparison of iPod managers
A selection of iPod to iTunes utilities:
TuneJack Windows Only (iPhone and iPod Touch compatible)
SharePod Windows Only (iPhone and iPod Touch compatible)
iPod2PC Windows Only
iDump Windows Only
YamiPod Mac and Windows
iPod Music Liberator Mac & Windows
Floola Mac & Windows
iPodRip Mac & Windows (iPhone and iPod Touch compatible)
iPod Music Liberator Mac & Windows (iPhone and iPod Touch compatible)
Music Rescue Mac & Windows (iPhone and iPod Touch compatible)
iGadget Mac & Windows (iPhone and iPod Touch compatible)
iRepo Mac & Windows (iPhone and iPod Touch compatible)
iPod Access Mac & Windows (iPhone and iPod Touch compatible)
TouchCopy Mac & Windows (iPhone and iPod Touch compatible)
There's also a manual method of copying songs from your iPod to a Mac or PC. The procedure is a bit involved and won't recover playlists but if you're interested it's available on page 2 at this link:
Copying Content from your iPod to your Computer - The Definitive Guide
I would also advise that you get yourself an external hard drive and back your stuff up, relying on an iPod as your sole backup is not a good idea and external drives are comparatively inexpensive these days, you can get loads of storage for a reasonable outlay:
Back up your iTunes library by copying to an external hard drive
