Bicycle Mechanics - New freewheel adjustments

I just replaced my 6-speed freewheel with a 7-speed. Now I feel like the locknut on my rear hub is set too far in, and the smallest cog is right against the frame - so it's like I replaced a 6-speed freewheel with a 7-speed that really only has 6 good speeds.

I'm trying to figure out what to do and wanted to bring it up here to get some ideas from all of you.

I figure I'll need some sort of spacer on the freewheel side to bring the locknut out, but what would that do with the rear wheel alignment in the frame? Would it need to be balanced on the other side?

Will I need a longer axle for all of this? And, if so, will I need to cold set my frame?

I know, a lot of questions. But I've never done this kind of thing before and am trying to get ideas now so that I can go into this fully prepared instead of having the bike on the stand for a week while I try to figure out what to do next.

Spacer works fine, if axle is long enough. You need at least half the width of the
dropout, better a bit more, on each side in axle length. Alignment worries are a
minor consideration especially as there is no reason to believe alignment is perfect
now. A bit of frame stretch on the order of 1-2.5mm is not a big deal either, more
would suggest the need for cold stretching, assuming a steel frame. You might
eyeball the existing spacers on R and L side, sometimes you can swap sides for
assymmetric spacers and shift the wheel over. That might necessitate a redo
on spokes to recenter the rim in the frame. Rim centering is more important than
a bit of stretching in the dropout spacing.

According to Sheldon, you don't need any axle in the dropouts for a secure wheel. What matters most is that the axle is not too long causing the quick release to clamp the axle instead of the frame. So, my point is that even with a 6mm spacer (assuming your frame was 120mm and the new 7 speed freewheel would prefer a 126mm spacing), you should have enough axle left to get the locknuts on securely which is all that really matters.

You will need to redish the wheel after adding this spacer though otherwise, your brakes will have to be way off center to work properly.

If it was my bike I'd move about a 5mm spacer from the left side of the hub to the right side. That'll involve redishing the wheel too. That's obviously a little more work but I think that it's well worth it to feel like you've done it right.

If it was my bike I'd move about a 5mm spacer from the left side of the hub to the right side. That'll involve redishing the wheel too. That's obviously a little more work but I think that it's well worth it to feel like you've done it right.
I agree with the concept but I think 5 mm is too much. I just measured a couple of bike (one Campy and the other Shimano) and the clearance between the outer face of the lockring and the inner face of the dropout is only 2 mm in both cases. Yeah, I realize the OP has a freewheel, not a cassette, but the concept should be the same.

I further assume the frame is already spaced 126 mm since it was 6-speed to begin with. So, I'd add a 2 mm spacer under the drive side locknut and see if that gives adequate clearance. The resulting 128 mm OLD should go into the frame with little problem. If that works, the required redishing should be minimal.

I found 1 mm was enough in a similar situation. I added two 0.020" washers behind the locknut on the drive side.

Yeah, usually 2mm is all you need. It really depends upon how much space the OP's rear-wheel needs. Adding 2mm washers on each side will move you from 126mm to 130mm OLD spacing without needing to re-dish the wheel.