Transmission fluid is black and smell burnt, and I think my transmission is slipping

I don’t know if the previous owner ever change the transmission fluid, but my car has 100k miles already. Today, I took the transmission fluid dipstick out, it look black and smell burnt, but I don’t see any metal. I don’t know how to take out the pan and check for metallic pieces, is there any auto shop that would do it for free?
When I accelerate to 40mph from a complete stop, the RPM goes up the 3000 and then quickly drop to 2000, then gradually drop to 1300. My car speed up really slowly even when I keep holding the gas pedal down with my feet. Is my transmission slipping? Sometimes when I start my car or when I change gear, my car jerk a little.
I’m afraid it too bad already and my transmission is depending on the broken metal pieces, so I don’t know if I should flush it? Will changing the transmission fluid fix the problem?

Remove transmission pan and check for metal filings for free. Absolutely not going to happen.

Changing the fluid may fix the problem,and it might not. It works sometimes. Since it is 30 times less expensive than replacing the transmission, I suggest you try it. If you can do it yourself, it costs maybe $30 to $40.

If the tranny fluid is burnt, take it to a shop that has a tranny fluid exchange machine and have ALL the tranny fluid replaced.

Then cross your fingers.

Tester

They also need to drop the pan and change the filter…Figure about $100 for this service.

year, make, model of vehicle??

I have resurrected more than one vehicle by dropping the pan and replacing the filter and fluid. It’s worth a try. If you do then repeat the same in a couple of weeks. Your transmission could recover from poor maintenance practices.

I think you’re going to be needing a new transmission. The only question is whether it’s now or very soon.
A lot of that black and burnt material is the friction material from the clutches; the rest of it fried fluid.

My car is a Toyota Camry 2003, automatic transmission. If I drive gently, how many more miles can I drive before I need to replace the transmission?
A local auto shop there charge $160 for change the transmission fluid, replace the filter, and clean the pan.

The question is not ( how far can I drive ? ) but do I want 80000 lbs of semi-truck behind me when it fails?