I can’t comment on the prices, but it does seem like you have transmission problems. And that price would be for a car with a decent transmission. You can take ti to the dealer and ask them for an appraisal. Take the letter with you. It is possible that they might pay you$4815, but expect a reduction for the transmission. BTW, you need not tell them about the transmission. They are experts and should be able to figure it out themselves.
One more thing: change the transmission fluid every 30,000 miles in the future. The transmission will function much longer without repairs if you do.