can i save my really cherished sms/imessages onto my computer????

SMS are saved in the backup, however you cannot access them directly. Look for a backup extraction program, or a 3rd party software like Touchcopy which can access SMS and save them directly to the comptuer.

I have made screen shots for a couple of messages I wanted to save.

You could get this computer program for you rcomputer called ibackupbot and extract the messages and then save them

Try the computer apps PhoneView (Mac) or TouchCopy (Mac & PC):

http://www.ecamm.com/mac/phoneview/

http://www.wideanglesoftware.com/touchcopy/index.php