Bicycle Mechanics - New front derailleur needed?

I'm fixing up an old road bike and I'm having some trouble with the front derailleur. I've got the rear shifting pretty well, but this one just doesn't want to change rings. Even when I get get it to shift up to the big ring, the chain will rub against the derailleur cage like it's not moving out far enough. Adjusting H limit didn't help much.
I feel like the spring isn't flexing enough. When I try to shift it, the cable gets very tight but the derailleur itself doesn't move very much. Any thoughts?

The derailleur must be cleaned and lubricated, not badly corroded and the cage needs to be straight with no chain wear/damage on the inner portions to work properly. If all that is OK then the derailleur needs to be at the lubricated and clean, at the proper height, rotated correctly, and limit screws properly adjusted. Check the Park Tools site and follow the entire procedure. The spring merely moves the derailleur back to the inner ring(s), has nothing to do with the shift up unless it is a Suntour that shifts inward when you pull back on the lever.

When you say "fixing up an old road bike" what, exactly does that mean?

Everything on a bike works together and every component has evolved over time. I can imagine a bottom bracket spindle that's too long, or an old, non-indexing front derailleur coupled with modern, indexing shifters. It could just be the adjustment too. Front derailleurs are a lot more picky about being adjusted properly and every adjustment that you do on a front derailleur affects other adjustments.

I once had a FD that was just toooooo loose in it's joints - did the same thing. And, it wasn't that old, and always keep properly lubricated. Shimano waranteed it.

When you say "fixing up an old road bike" what, exactly does that mean?

Everything on a bike works together and every component has evolved over time. I can imagine a bottom bracket spindle that's too long, or an old, non-indexing front derailleur coupled with modern, indexing shifters. It could just be the adjustment too. Front derailleurs are a lot more picky about being adjusted properly and every adjustment that you do on a front derailleur affects other adjustments.

It's a Shogun Selectra, not sure what year exactly. Downtube shifters, front seems to work on friction, rear is indexed (got that one shifting smoothly). Whole group is marked Shimano SIS. Not sure about bb spindle length, I could check on that later.