Recent grad with bad credit in need of auto loan

Yes, showing your offer letter can help you get a loan, from both credit unions and dealers. It's especially useful as a way to provide evidence of a raised income for someone who already has decent credit. Some lenders will require seeing a couple of pay stubs, while others can be satisfied seeing the offer letter before employment starts. Of course, given your credit history they'd likely be on the cautious side - which might mean rejection or unfavorable terms.

You probably do have some options though. If you can get by on public transport for a while, try to do that. If, as you say, the local transport options are awful, think about getting a beater off Craigslist or something. It doesn't need to last you more than a few paychecks. Borrow from family if you need to. Or see if you can carpool with a friend or a colleague. If that's also not an option, try a good credit union first. They're probably more likely to reject you than a predatory "bad credit welcome" dealer, but they often have an interest in helping their members improve their economic situation, and might be willing to take a chance on you and a small used car loan. If all this fails and you get desperate, well, that's not a good negotiation situation to be in...

As for trying with random lenders and hurting your credit score, multiple hard inquiries in a short time span typically get grouped. So shopping around won't hurt your score much, if you do all the shopping around at the same time. Still, there are many pre-approval lenders (including credit unions) that can do a soft inquiry and give you an idea about whether you're likely to get a final approval or not. Maybe start with those to get an idea about it, as it shouldn't hurt your credit score at all.