The refresh rate of your monitor will increase the capped FPS when v-sync (or "sync every frame") is enabled, however this cap is not causing your poor FPS.
A more powerful graphics card will help smooth these situations; a 5770 is not a very powerful card, because COD games are generally a broadly marketed game on both PC and Consoles, they have a lot of optimizations (and are not extremely demanding graphically to begin with I hear), so a lot more effort has gone into getting that game to run smoothly. Need For Speed games, on the other hand, are popular on consoles, and most of the PC versions are minimal-effort ports. A more powerful GPU will help many situations where your framerate dips, but it's not a silver bullet either (as all the other stuff above can also cause hiccups even with powerful graphics and CPU, sometimes it's just the game).
A new TV will not impact your minimum framerate (which is the root of your complaint). A new GPU will definately improve your average FPS, and should resolve most of the stuttering you would see.
A 7870 is a pretty strong choice for 1080p@60hz it's powerful enough to run today's games smoothly at max settings, and should be able to run high detail levels for several years.
Keep in mind that when upgrading your GPU, the capability of your power supply is very important in determining what graphics card you can get, often prebuilt PC's have a PSU sized only for what the computer is sold with, so it maybe neccessary for you to upgrade that as well. For a 7870, a 500W power supply is recommended, if you have any concerns, be sure to post your power supply model, rating and the Amperage of the 12v rail(s). If your PSU isn't adequate, it could damage the PSU (and possibly the system) and will cause instability during gaming (and other demanding situations).