I have a radeon 5770 as well (not the same aftermarket manufacturer, but same card for all intents and purposes. First to clarify, you're not getting "lag".
Some things I would look at given the description of your problems, first make sure that in CCC/Ati Tray Tools you do not have frame buffer set to 1. This will generally cripple the performance of the 5770 on modern FPS's on a low-end system like yours (I also have an old dual core setup so I feel your pain). I prefer this set to 3, and in bad company 2 I used a frame buffer of 4 which actually made my highest overclock quit occasionally causing driver failure when scoping in and out quickly.
Another thing that you should look into is try ensuring that VSYNC is disabled. I say this only because in some cases, particularly with in-game set vsync, the vsync sets a standard divisor and if your frames drop below 60 (or 90/120 in your case with your CRT), it does not vary the divisor, leading to choppy gameplay and some excessively low fps (commonly in situations where you would have had 45-50 fps without vsync, the aforementioned glitch can drop you to the 25-30 range).
Look into overclocking your cpu and 5770 (and for this I recommend ATI Tray Tools since it does not have a top-end limit on the overclock slider, and the 5770 can surpass the upper-end 960mhz core clock that CCC tries to restrict it to). With my 5770 in most FPS online multiplayer I play at clocks usually 1010/1240, 1000/1300, 1010/1200. It depends on the game as to what clocks will remain stable on stock voltage. ATI Tray Tools also lets you set the voltage to constant max voltage allowed by the bios, instead of having the power-saving features enabled and this tends to make higher overclocks remain stable in games without having to flash to another bios.
This may sound dumb, but make sure all of your connections are tight and that your 5770 is in the correct lane on your motherboard. Check even the connection at the base of your monitor as well as the back of the computer.
In general, in Deus Ex Human Revolution, with my 5770 and dual core setup (and 4gigs of DDR2 ram at 800mhz lol 🙁 ), I'm able to maintain 50-60 fps fairly easily at 1680x1050. Make sure that you do not have things like anti-aliasing (AA, MSAA, etc) enabled as well. The only drivers that work well for a dual core cpu/5770 setup are 11.6, 11.6b, and 11.9. Make sure you are using one of these driver sets.
Your monitor being a CRT has nothing to do with low framerates as the above poster suggested and CRT's have for a long time been generally renowned as the preferred monitor type of some of the FPS-genre's elite players because of the resolutions they tend to come in, as well as their high refresh rate without causing image blurring (which generally occurs in most 120hz LCD's).