I have done both.
Let me start out by saying I have a relative who owns an auto repair shop in San Antonio so I might be able to get you some inside advice. Feel free to PM me.
You are always going to do better financially if you sell your car. It is also going to be a lot more effort if you do a private sale, and you have to deal with scam buyers*. Even with legit buyers you have to ensure that the form of payment is valid (take cash), coordinate to transfer the car and title in such a way that they can register it (meet at DMV), etc. I have used Craigslist, Autotrader, and Facebook Marketplace. The number of scams has gone up over the year since the first time I did this because the scammers’ automation is getting better.
It’s pretty easy to sell it to a dealer like Car Max–you will save a lot of hassle but get less cash. However, with the damage you described, mechanical problems (including unknown ones), and high mileage (more than double the average), they may not take it. If they do, they’re going to lowball you. There might be local used car lots that would take a car like that but I don’t know, I’ve never dealt with one. Also a lowball situation.
Donating is easy, they’ll take anything. However, your financial advantage could total zero. With the current tax code, most people are better off taking the standard deduction rather than itemizing, and you can take a charitable deduction only if you itemize. If you itemize, find out from the charity if you will be able to deduct full market value, or something else like an auction price. I donated a minivan to a local high school’s auto mechanic vocational program. This was before the current tax code, so I was able to deduct it. For this program, I was able to deduct the full fair market value of the car, because they were going to improve the car then sell it. The school gave me a tax form recording the donation and listing the deductible value.
I priced out your car on KBB (using standard equipment, Fair condition, zip code 78015) and it came in at the range $1,425-3,128 for a private party sale. Your car is probably worse than Fair condition, so that price is optimistic at best. The bottom line is 1) how much work are you willing to do for that amount of money, and 2) what is the value to you of making a charitable donation.
*The most common scam is that they tell you they are out of town, but will buy the car sight unseen for your asking price. They will send a transport service to pick it up. They will send you payment in a cashier’s check for the price of the car plus the transport fee. They will ask you to wire the transportation fee directly to the transport company. By the time you do all of that, a week later the cashier’s check bounces because it was a counterfeit, and you are out the transport fee with no way to identify the buyer or recover your loss.