The AMD HD 6770M is a version of the AMD HD 6770 intended to be integrated in to a computer logic board by the computer manufacturer, as such it would only be used in a laptop or an all-in-one computer like the iMac. It is therefore impossible that you have one of these in a classic Mac Pro which only uses PCIe card based video cards, i.e. the AMD HD 6770. The 'M' at the end can be considered to refer to 'mobile' as in laptop.
I will therefore assume you actually have an AMD HD 6770. (Without the M.)
This is what the AMD HD 6770 would look like - http://www.amd.com/en-gb/products/graphics/desktop/6000/6770
This is what the AMD HD 6770M would look like - https://www.techpowerup.com/gpudb/2254/radeon-hd-6770m-mac-edition
Note: The 6770 and 6770M are in computer terms ancient designs since they date from 2011.
Apple has used the 6770M in a MacBook Pro dating also from 2011 and also an iMac again dating from 2011. The MacBook Pro with this 6770M chip had a known design fault that resulted in Apple offering a free warranty repair extension program for this issue. See - http://www.apple.com/support/macbookpro-videoissues/
The 6770 (without the M) was as far as I am aware never sold as an official Mac edition. This means it did not have Mac firmware meaning it would not display anything before the Mac finished booting. The 6770M as used in the MacBook Pro and iMac obviously did have Mac firmware.
The drivers for any and all AMD cards are not written by AMD but are instead written by Apple. The only updates - if there are any would come with a newer version of OS X. You cannot download drivers for the Mac from AMD.
If you have a PCIe version in a Mac Pro and not the MacBook Pro I suggest either your cards fan might need cleaning, or the heatsink compound for the main video chip on the card might need replacing since your issue sounds similar to overheating type problems. You could consider getting a newer (in relative terms) card e.g. the HD 7950 or since you are using Adobe software you would in fact be far better off with an Nvidia card instead.