A bumbo chair or other bath specific chair will help you if your baby is sitting by themselves. If not, bathe that baby in the sink instead!
I’d say purchase a shower chair for her or if ur in with her put a towel on ur leg or around her and sit her there so she’s not slippery
I totally get that would make you paranoid, but accidents happen mummy and you caught her super quickly. Newborns are bloody hard to bath because they slip, slide and flop everywhere so please dont punish yourself ❤️