For $350 you're better off using your smartphone (assuming you have one) and maybe looking for a cheap gimbal which will take your video skills from abysmal to passable with less than a week's worth of practice.
I might also suggest trawling through flea markets for low cost film gear.
Or if youre insistent, a second hand 450D might be found for around $300-350 with a 50/1.8 but there's no fucking way you're going to find anything even halfway capable with an APS-C sensor or larger that can also shoot video for anything under $500