I got accepted into my dream program at one of the best universities in the US, and I'll be working under an excellent, very well known advisor whose interests align very well with mine. However, since I will likely be getting a multi-year fellowship, and due to how this specific program is structured, it seems that I won't be getting much TA experience, if any at all.
I love teaching and enjoy it very much, and I already have 2 years of TA experience as an undergrad. How much would it affect my future TT applications if I have a (hopefully) solid research record but not that much TA experience? I don't want to turn down this opportunity but might have to if it turns out I won't get the TA experience I want/need.
EDIT: My field is applied physics/electrical engineering/materials science (very interdisciplinary research area so in theory I could be hired by any of these departments). I hope to work at an R1 school.