6500 series and 7700 series STI are 9 speed levers.
BUT
They are different internals. Not hugely but no, you can not interchange the internals.
I have taken both apart and overhauled both as well as the 105 models. In 8, 9 and 10 speed. Have not done an 11 speed set yet but I have done the others with great success.
The whole reason for overhaul is the Shimano grease. It gets hard over time. solid hard. And they need degreased/degunked and relubed and reassembled. If you pay attention when doing it they are hard but manageable. Just pay attention to everything and keep an eye on the small parts.