You don't need the dashboard to confirm whether or not the game saves are there if you can cold boot with the game already in the drive when you turn it on and then check to see if the game lists the saves, but I'm guessing that as of now, cold booting still results in an error 16? I know that whufclee already mentioned trying to force an error 12, but just to confirm, you tried to boot the Xbox with the IDE cable disconnected from the DVD drive, correct? And if you did, then what error does it give you? If it's error 12, then the hard drive should be unlocked and you should be able to hotswap it.
If hotswapping doesn't work, then your other two options are to either install a modchip (at least temporarily to fix everything) or build an
EEPROM reader. There are solderless modchips but they're not only hard to come by nowadays but they're probably more expensive than you'll want to pay. There are cheap solder modchips if you're ok with soldering. The EEPROM reader will require soldering to both put it together and to install it into the Xbox, and it could be cheap depending on how you source the parts, and installing it will be about the same level of difficulty as installing a modchip.
I actually have a solderless modchip that I'd be willing to let you borrow if you'll pay for shipping both ways (I'm in California) and leave a deposit with me via Paypal that I'll refund once I get the modchip back. The modchip is an Xecuter 2.3 Lite+ (or Lite Plus) and it's only good for Xbox versions 1.0-1.5, so if you have a 1.6/1.6b, then it won't work for you (at least not without soldering). Here's the
basic installation guide if you're interested (the guide seems to leave out connecting the power splitter but I can talk you through it). If you want to do this, then leave a message here and also send me a PM and we'll work out the specifics, and I'll also test the modchip to make sure that everything works.
In any event, don't trash the Xbox because it's still fixable. At the very least, offer it for free on craigslist and I'm sure that someone will take it off your hands.