I've taken three 6 speed downtube shifting bikes to 7 or 8 speed STI brake/shifters in the last year. The 8 speed bike got respaced to 130 mm (potentially dangerous modification) and a new cassette wheel (8-9-10 compatible). The other six speed bikes got a new seven speed freewheel for their existing wheel, and kept their 126 mm dropout spacing.
The 6-7-8 speeds are all spaced differently, and require different shifters to pull different amounts of cable for proper indexing. I recommend going for the 7 speed freewheel. FYI, if you are hard on axles then the wider 7 speed freewheels can be worse than 5 or 6 speed setups.
<$10 for freewheel removing tool. I mount the tool in a vise and turn the wheel.
Nashbar has 7 speed 13-15-17-19-21-24-28 or 32 freewheels for $10 to $20. A 32 tooth low makes a nice bail out gear and worked ok with 42/52 front chainrings and a short cage 105 rear derailleur. A new chain was installed to wrap over the 32-52 big-big combination. You may need to re-dish the wheel and respace the hub to get the best fit. This requires cone wrenches and a spoke wrench.
You will also need: compressionless shift housing, new shift cables, downtube cable stops, and the shifters.
Older RSX left brifters won't trim the front derailleur. Newer "plasticky" Sora 7 speed brifters worked well for me, and the left will trim if needed. A triple left brifter will shift two or three front rings ok.