well in addition to the fact that my alternator sucks just like yours thats not the problem i have the same exact problem like just about everybody else and i took it everywhere and only one dude could figure it out toyota shop told me i needed a hole new exhaust but another shop lifted the car and showed me
i let a crappy shop weld my catalitic converter and while they were hammering it off they busted my flexpipe so just get a new one cus i think its in front of or some where near a sensor and they said that was fucking up my ecu so check your exhaust system im pretty sure thats whats wrong with mine