I had lots of sentimental messages as well on my phone and I simply played them on a speaker phone and recorded them to a tape recorder. Now I have a tape of all the messages I wanted to keep. If you can't play the messages on the cell phone directly if it is broken, are you able to call into your voicemail using another phone with a speaker and play them over the speaker while recording to a tape recorder? That's the easiest way and then you have a tape to keep forever.
If you want to get a great deal on a new cell phone and a new service plan, check out the website below .They have lots of very cool FREE cell phones with a new service plan from any of the major carriers. You can compare all the carrier phone plans and get a great deal on a new cell phone.
if they are voice messages i would call voice mail then do three way calling to someones or your house phone and let the answering machine record them.
if the are text messages i really don't know how to save those except copy them on paper and have someone artsy do a whole page with pictures and the messages framed nicely.