I get drunk easily, even infrom small amounts (of alcohol).

Either "from" or "with" would work.

I get drunk easily, even infrom small amounts of alcohol.

Maybe it wasI became tipsy, maybe because I cooked ithe chicken slowly at a low temperature, so the alcohol wasdidn't all over the place, and I became tipsyevaporate.

Personally, I think the sentence makes more sense if you put what you are about to explain at the beginning. Instead of evaporate, I think "cook out" or "burn off" can also work here.

To keep it the order you wrote: "Maybe because I cooked the chicken slowly at a low temperature, so the alcohol didn't evaporate, I became tipsy."

If the "alcohol wasn't all over the place" is there because you want to express why you cooked it slowly I might write: "Maybe because I cooked it slowly at a low temperature, the alcohol didn't go all over the place, but the un-evaporated alcohol made me tipsy."

I get drunk easily, even infrom small amounts.

Maybe it was because I cooked it slowly at a low temperature, so the alcohol wasdidn't all over the place, andevaporate, I became tipsy.

I think by "all over the place" you mean the alcohol did not disappear, right? "Evaporate" is the scientific term. A more common choice might be "cook off": "so the alcohol didn't cook off".