Many houses built in the era of yours did not ever receive a finish coat of plaster. It was assumed that wall paper would always be used. If you remove a bit of the final mystery layer, what's underneath?

Try a bit of sanding, either wet or dry where the 'seams' are...if that works, you have your answer. :) On one section of my stairs, the painted paper was cracked, so I removed that section, and revealed what looked like sanded surface. I skim coated that section, and removed all the paper in the downstairs foyer and the rest of the stairs--didn't see that in any other spot, so I think the finish coat had come off with the paper in that spot.

The wallpaper also often leaves stains on the wall even after it is removed.

Use Easysand or Durabond to repair the damaged finish coat unless you have a source of lime putty.

Durabond is so hard it is very difficult to sand.
Easysand is just that, easier to sand.

Both are setting type joint compounds.
You should use a bonding agent on the areas needing repair.
It prevents the old plaster from absorbing water the new material needs to cure correctly.
EUCO WELD is a good brand.
If you cannot find plaster bonding agent use Elmer's white glue cut 50% with water.

In either case, paint the agent on and wait for it to get tacky before putting the patch on top.

Underneath the white layer is grey/brown plaster. The white coat is thick and rigid. It's probably between 1/8 and 1/4 and inch. It's really smooth (on the flat parts).

I'll try sanding the "seams" to see if that will help, or give me some idea on the next step.

Thanks!

The gray/brown layer is the 'brown coat', which was either papered, or had the 'finish' coat put over it. Since your white coat came off in places, that was the fnal one. Patch the spots where that happened, and you'll be all set to repaint or paper.
Old plaster was almost always three coats: 'scratch' was reallyrough and is what holds to the lath, 'brown' goes on next, and is smoother--suitable for paper, 'finish' is last and can be painted directly or papered.

I'll try washing the walls. The lines are exactly where the paper was, so I'm thinking it might be residue or some sort.

Thanks!