As far as I know voice-mail messages reside on your carrier's server and not on your handset so there isn't a file on your phone to recover.
Can you call the voice-mail number from another phone and then, whilst retrieving the messages, record the call to an mp3 with the '
Record' feature of the phone you're using to make the call?
I'm not too familiar with Sprint's voice-mail setup, as I'm in England, but I know that with my carrier I can access my voice-mail by dialing my own mobile (cell) number from any phone, land-line or mobile, and then hitting * after the tone and entering my PIN.
If you haven't already setup a PIN in your voice-mail, and you can't access the dialpad on the broken phone to set one up, you should be able to make the call to voice-mail using the broken phone and Bluetooth, as you said, and then use the keypad tones from another phone to access the menus and setup a PIN.
That's if Sprint Customer Service won't help you with the PIN setup of course!
Once the PIN is setup you should be able to use any phone, or even a PC and Skype, to access your voice-mail.
In fact, come to think of it, a PC and Skype is probably the easiest way to access the voice-mail 'cus then you have the call coming through the PC and you should be able to record it right there on the computer.
Also, are you sure that the touchscreen doesn't work? When the display on my S2 failed, the touchscreen was still fully operational and I was able to access the full functionality of the phone, once it was connected to the PC, using a great bit of freeware called
MoboRobo. MoboRobo enabled me to see what should have been on the display and then I just used the phone via the touchscreen.
Hope this gives you some ideas; good luck!