One year's salary as a down payment?? Yikes. At 17K a year, unless you live at home and this isn't your "real job" (aka: you're just making some side cash while going to college) don't buy a 24K car.
Calm down, no one is judging you. I didn't see that you were a college student - saving up 16K as a student is great and getting a car for 24K is fine. I thought you were a post-college adult making 17K per year and wanted to use 16K as a down payment - that seemed like a bad idea to me. In this community, if you posted information and want answers you should expect members to sometimes ask more questions or give advice. Don't get so riled up when we do.
To answer your question, you don't need to specifically go to a first time buyer program. I just applied with BoA as a first time auto buyer, no mortgage, etc. and was approved with their best rate online instantly. Try your local credit union too.
I recommend a trying your local credit union. Or even trying to join one like DCU. They may be more lient that tradition venues.
USAA just approved me at a great rate as a first time buyer.
Subaru thinks they can offer me their 0% rate, and they did not mention the 50% income restriction.
The NIssan dealer by me was offering a Leaf for $129/month lease.
That might be something for you to look at. You would get to keep that money you have saved up, plus you would save money on gas. As long as you don't have to drive long distances then it could work well for you. Keep the Camry around for road trips and use the Leaf as a daily driver.
I can not see why you won't get approved. Over 50% down, the finance companies will love you. Don't extend the repayment to long, but not too short either. You can always prepay your loan if you have extra money. Good luck, time is your best friend. You are just beginning your credit journey and doing very well. P.S. you did not mention if you still had emergency reserves. Probably not good to empty the savings with the current low interest rates. Also be sure to have enough cash to pay for sales taxes, licenses, and one year of vehicle insurance.