I agree with the professionals who suggest you speak to a lender. But, here is the real question....Where do you feel comfortable with your overall payments? True, eliminating the car payment will get you more purchasing power, but you may not need that. The $3000 in a down payment, to reduce the monthly mortgage, only saves you $18 per month. Its $6 for every thousand borrowed at 6% interest rate on a 30yr fixed. (Adjust for higher rates, but this is a good barometer.) When you look at the 2 options, $3000 put toward saving $20 per month, or saving $317 per month is a no-brainer. Still talk to a lender as they also have trade line requirements, on top of credit score, income/debt ratio, job history, etc...requirements.
Just don't let many of them run your credit. Let the one you like the best run it, for now. Your middle credit score is what lenders will use to educate you, on what programs they have for you. Sounds like you may just go with a conventional mortgage, but the PMI (when you put less than 20% down) can be costly too. Talk to someone you trust soon, to get you on the right path.
I'm not licensed in CA (and its a far drive from Philly) but if you have any questions or need someone out there, let me know. Visit my web site for more useful info too.