1. Assuming that the rifle has 3 screws for the bottom metal, the front and rear screws should be tight, ~50 in/# or so in a wood stock. That middle screw needs to be just finger tight enough to not back out. If you make that screw too tight, you'll flex the action, accuracy and feed/function will both be wonky. If you pull it apart, when putting it back together it's front screw 1st all the way to "tight". Rear-most screw next ~3/4 tight. Middle screw, tighten until the floor-plate operates correctly, then tighten the rear screw the rest of the way.
2. Assuming you have rings and bases, take the scope/rings off the rifle, leave the bases in place.
3. Remove the rear screw from the front base, try to wiggle the base with only the front screw holding it down. On 2 occasions, I've had that front screw be too long, contact the bbl tenon so it felt like it got tight, but didn't actually get tight against the base.
4. For the other 3 base-screws you can use your pinky finger to feel if they are coming through or not. Most likely not as they would all come in contact with the bolt if they were too long. If the rear screw of the front base is too long, the bolt won't cam down completely. If the screws are too long for the rear base they'll drag on the bolt body.
5. With the bolt in the firing position, tighten down the front action screw, now try to open the action, if the bolt won't open, that front screw is too long. If it's too long you either can't get it tight enough and have to leave it a little loose, or it's snugged right up against the bottom lug of the bolt which would again cause accuracy wonkiness.
Should you decide to re-barrel, most aftermarket bbl makers have the Winchester Fwt pattern available. You'll be able to choose what twist you'd prefer within reason. Figure the job to be +/- $700 for the bbl, chambering/threading.
Given where current projectile R&D is going, if ordering a new bbl, I'd order 1:7 or 1:8. Take your pick on a bbl, everybody builds a good 6mm tube. Were it me, I'd probably just order whosever I could get in a reasonable timeframe. The last bbl I ordered was a Lilja to duplicate a Kimber MT factory bbl and it took 8-10 weeks from order to delivery.
You mentioned Featherweight. SS or blued? Push-feed or "Classic" controlled-round feed? New Haven, South Carolina, or Portugal production?