Or perhaps I should try to get away from my web developer job and get into something else?

You already know what to do here.

I just quit my job at MSFT sometime back, worked straight outta college for four years (in the MS-IDC office). I quit to actually pursue Research in "systems". As for you not getting a callback, add the right keywords/buzzwords in your resume and most importantly LC (considering where you are in your career LC is more than enough). Thats all there is to it. You have a Masters degree from US and you seem like a curious person, I have seen people with far less get into a FAANG job (in India at-least).

FAANG job would definitely be a step up in all aspects money, career etc, for a couple of years at-least. But I get a sense maybe you will grow out of it too (just like I did).

Good luck with whatever you do.