I answer this question as an AI person, because this is a real problem of natural langues (like English, French, Persian, ...).
First of all your answer is : Yes, the sentence is ambiguous, like many other sentences which we use in our every day conversations. It's indeed intrinsic to so called natural langues. And let me don't agree with the other answer, cause I think the right answer should be given according to the context from which the sentence is taken.
e.g.: "Ok, Let's play football before watching TV". Who wants to watch TV? Look at these sentences that me appear prior to it.
1.B) "I want to watch TV" -> B wants to watch TV and A asks him/her to play
2.A) "I want to watch TV, but" -> A wants to watch TV but accepts to play
3.C) "Let's watch TV" -> C invites them all to watch TV, and the accept but after playing.
I didn't try to create the sentences very carefully but I think they're enough to achieve my purpose.