Probably under the Skrulls custody. I doubt he's been one for long. I don't think the president thought he was acting strange (drunk) just because of his/her breath but because he was acting different than usual. Same way Nick figured it out.
(Side note but it was interesting that a female Skrull has been portraying him. For some weird reason, so far male Skrulls have only impersonated human men and female Skrulls human women. Which is crazy considering they can shape-shift into pretty much anything )
One thing I find odd that hasn't been brought up is how the Skrulls haven't been given the home that was promised, yet the Asgardians were welcome to Earth and given a place for their own in Norway where they can be themselves. You'd think they'd be pissed about it. (Then again, New Asgard seems to be a small population unlike the at least 1,000,000 Skrulls walking on earth).