I'm pretty sure you will have to replace the entire motherboard to replace the graphics card (and it does sound like the graphics card is the problem). That's going to cost a lot -- probably $500-800 if you go through Dell. eBay may be slightly cheaper, but doing a motherboard replacement yourself is not for the faint of heart.
Computer: Intel Core i5-750 2.66 GHz quad-core processor | Asus P7P55D-E motherboard | Crucial 4 GB DDR3-1333 RAM | nVidia GeForce 7800GTX | 2x WD Caviar Black WD1501FASS 1.5TB hard drives in RAID 1 | Antec Sonata III case with Antec EarthWatts 500-watt PSU | Dual Dell UltraSharp 2408WFP 24" widescreens | Windows 7 Ultimate 64-bit
Other: 2007 Subaru Legacy 2.5GT Spec.B 6MT | Samsung Galaxy Note II Smartphone | Mamiya M645 1000S medium-format SLR with 55mm f/2.8, 70mm f/2.8, 210mm f/4, teleconverter, 120 and 220 film backs | Olympus E-PL1 Micro-4/3s DSLR with 14-42mm and 40-150mm lenses