I'll step in and advise that 2GB of vram is a _minimum_ upgrade these days for smooth gaming, especially towards the upper end of the spectrum, which 1080p qualifies as.
You said don't break the bank. Might I suggest a gtx 760 with 4GB? Should run around $300. Or a 770 which will set you back another Benjamin.
After these, your next step up is a plain-jane GTX 780... which is what I'm currently using at the 1920x1080 resolution you're targeting. And yes, it works great. Figure I've got another two years out of this card minimum, assuming Star Citizen doesn't eat its lunch. (it's a little sluggish atm, but I assume tuning will bring performance up as they go)
Gaming res is increasing. Game texture resolutions are increasing. Stuff like post-processing and AA are getting more intensive. (yes I noticed you mentioned no AA atm)
Thing to remember folks... if you want the nice screen and the big game textures... the gpu you need to drive it at decent framerates starts getting pretty expensive