sync with a new library and not lose songs

The easy way? Copy your iTunes library (the whole iTunes folder) from your old computer to your new one and add the tracks currently on the new computer to the old library. They you are syncing with an updated version of your old library. If you can't do this and you failed to maintain a backup then:

Some of the information below has subsequently appeared in a document by turingtest2: Recovering your iTunes library from your iPod or iOS device - https://discussions.apple.com/docs/DOC-3991

Your i-device was not designed for unique storage of your media. It is not a backup device and media transfer was designed for you maintaining a master copy of your media on a computer which is itself properly backed up against loss. Syncing is one way, computer to device, updating the device content to the content on the computer, not updating or restoring content on a computer. The exception is iTunes Store purchased content.

iTunes Store: Transferring purchases from your iOS device or iPod to a computer - http://support.apple.com/kb/HT1848 - only media purchased from iTunes Store

For transferring other items from an i-device to a computer you will have to use third party commercial software. Examples (check the web for others; this is not an exhaustive listing, nor do I have any idea if they are any good):

- Senuti - http://www.fadingred.com/senuti/
- Phoneview - http://www.ecamm.com/mac/phoneview/
- MusicRescue - http://www.kennettnet.co.uk/products/musicrescue/
- Sharepod (free) - http://download.cnet.com/SharePod/3000-2141_4-10794489.html?tag=mncol;2 - Windows
- Snowfox/iMedia - http://www.mac-videoconverter.com/imedia-transfer-mac.html - Mac & PC
- iexplorer (free) - http://www.macroplant.com/iexplorer/ - Mac&PC
- Yamipod (free) - http://www.yamipod.com/main/modules/downloads/ - PC, Linux, Mac [Still updated for use on newer devices? No edits to site since 2010.]
- 2010 Post by Zevoneer: iPod media recovery options - https://discussions.apple.com/thread/2452022?answerId=11624224022#11624224022 - this is an older post and many of the links are also for old posts, so bear this in mind when reading them.

Syncing to a "New" Computer or replacing a "crashed" Hard Drive - https://discussions.apple.com/docs/DOC-3141 - dates from 2008 and some outdated information now.

Copying Content from your iPod to your Computer - The Definitive Guide - http://www.ilounge.com/index.php/articles/comments/copying-music-from-ipod-to-co mputer/ - Information about use in disk mode pertains only to older model iPods.

Get Your Music Off of Your iPod - http://howto.wired.com/wiki/Get_Your_Music_Off_of_Your_iPod - I am not sure but this may only work with some models and not newer Touch, iPhone, or iPad.

Additional information here https://discussions.apple.com/thread/3933661?answerId=18324797022#18324797022