Used or refurb are going to be your only options in the present generation; you're a bit late for clearance new Haswell models, but that would be another.

Dell Outlet doesn't tend to have a ton of machines with the better resolution screens (which tend to be the IPS ones ) but have definitely seen machines with FHD IPS screens in that range.

They've got some very good sales now, but nothing reasonably small with a FHD screen in your price range. OTOH, you can get a Latitude 3340 with an i5 for $325 (plus tax if applicable in your state, ground shipping free) even less if willing to take a Celeron or an i3, or willing to go to a 14".

For disposable ones, Newegg typically has some nice older-generation Lenovos really cheap. Not sure if they have any better screens.

I've never had much luck getting good prices on the Lenovo site, and it's a very light 14" rather than a <= 13", but right now they have a T450s refurb for $515, with i5/8gb memory and an i5.