Mama, I know that worry, it's so scary pag ganyan ang temp ng baby mo, especially pag first time! But since your pedia already checked her and the tests came back normal, I'd really trust that professional advice kasi siya ang nakita talaga si baby mo.
A few of us moms have been in that alangan place too with antibiotics, but honestly, kung binigay ng doctor, may reason siya, maybe she's worried sa risk or there's something na kailangan ma-prevent. Best talaga is to ask your pedia directly why she prescribed it, and if your gut's still unsure, you can always get a second opinion from another pedia, but don't skip it without asking first, ha? The fever at the drooling sound like it could be teething, but your doctor knows best.
Gabayan mo lang siya exactly per prescription and watch si baby mo, you're doing great, mama.