Not a mechanic, but my dad is one, and my ex had an 03 TL. The Hondas from that era are plagued with transmission problems, specially the V6s and minivans . A Google search will show a lot of people with transmission problems on Acuras from that gen, and I even remember stumbling across a website dedicated only to gathering enough data/signatures or whatever to try and get a class action lawsuit goinf, and IIRC Honda did give a warranty on those vehicles transmissions, at least here in Canada

In the case of my ex's TL, the car would increase Rpms like crazy between 1-2nd on cold mornings. the dealer did a flush and topped it up with the Honda Trans fluid, which didn't do anything to fix the problem. They also did some checks (solenoids?) but everything seemed to "work". Hope that helps