Hi, a single HD6850 will sufficiently run both of your new monitors at their native resolutions in a desktop environment…not so much for gaming.
Regardless of the specific output ports you have on your HD6850, only the Dual-link DVI (one of the DVIs) and DisplayPorts support resolutions beyond 1920x1200, max 2560x1600 for both. If you have a VGA port then it supports max 2048x1536. This is mainly specific to older cards since newer revisions of both HDMI and DisplayPort support 4k resolutions.
That explains your first problem since the monitor you described as working properly was connected to a Dual link DVI port while the other was being limited by either the Single-link DVI, HDMI or VGA port you were using. Your new monitors, U2713HM, have DisplayPorts so connecting the second monitor directly to one of the DisplayPorts on the GPU should alleviate your problems. Depending on the type of DisplayPort connector on your GPU (regular DP or miniDP), get the appropriate DisplayPort Cable (DP to DP or miniDP to DP) since the port on your monitor is a regular DP.
If you don’t use this setup for gaming, I see neither a reason to shell out money for a new card nor even keep the second HD6850 you bought since a single HD6850 will do the trick taking into account the above. Otherwise for a smoother gameplay experience the bump in resolution could definitely benefit from extra GPU muscle so keep the second HD6850 and crossfire them while still connecting both displays to the primary GPU only as outlined above or buy a new single high end GPU.