This is a very debated topic. In technicality, if you use your TPS EAD you forfeit your F-1 status. But until your DSO knows, your SEVIS would not be terminated. If you want to be clean, what your DSO suggested is the best path. Regarding Green Card, this is a very subjective matter. However using your TPS EAD is not illegal so I do not see why it would be a problem for green card. However, I am no lawyer. I would do what your DSO said, be patient. However, if its an absolute emergency then do what you gotta do.

TPS EAD processing is definitely taking time. Almost 6-8 months. However, some people are getting it in literally 2 weeks too. Its weird. Unfortunately, there is not much you can do about it.