The source of leaks can be very difficult to track down. Water runs downhill, except when it is between layers of material. Then it can seep uphill, sideways, through other layers, where it then can drip to another surface to repeat the process almost indefinitely. Leak paths are routinely more perplexing than the Warren Commission's Magic Bullet Theory.
A photo from the top of the roof would be very helpful. But in general, walk the roof and look for the leakiest-looking areas and replace the shingles. Then look for gaps, cracks, and holes and fill them with roofing goo (this one can be used while it is raining).
Carefully scrutinize all valleys for cracks and punctures.
A common source of leaks is through-the-roof devices. Check specifically the seals on the side and top vents for water entrapment. For example, this photo highlights problem areas:
![chimney trapping water](https://i.stack.imgur.com/Wtaza.jpg)
Sealing around through-the-roof devices should be done with the roof dry and using silicon caulk. Even the goo on there must slope away from the fixture, otherwise it creates a "leak reservoir".
I once had a leak like this, which didn't look like it could be source of the near-waterfall underneath. But it was.
![leak around vent pipe](https://i.stack.imgur.com/bZ9v5.jpg)
Lead makes a good roof, but finding leaks in one (or any flat roof) can be a real bear as the leak below may be a long way from the leak above.
If the folded seams are (as I suspect from your description) standing up, except where walked on and flattened, then yes, the walked-on parts would be prime suspects for leakage. Sometimes the the seams are all folded down and soldered, but standing seams are not usually also soldered (it's a fair bit of fiddly torch work.)
Look very carefully for cracks right next to the flattened seams. If you don't see any cracks, run a torch over the flattened area (trying to anneal the metal before reworking it) and carefully stand the seam back up - look for cracks on the lower side as well. If there are cracks you (or a lead roof specialist you hire) can clean the area and solder them.
If there is a wall above the flat roof area, also look very carefully at the flashing where the wall joins the roof - likewise look very carefully around any penetrations in the roof.
Some folks will no doubt tell you to reroof, since lead is "old-fashioned" and therefore also "old" - depending on the shape the roof is in this might be the right thing to do, but both lead and terne-plate (lead coated steel) roofs can last for hundreds of years, so it is well worth trying to fix it first, IMHO.
Best Answer
The two main reasons I know of for condensation in an attic are a lack of air circulation from soffit to ridge vent and air leakage from the house typically due to an improperly vented exhaust fan or insufficient vapor barrier between the insulation and top of the ceiling.
I'd focus on the first cause by making sure your soffits are vented and there's a clear path from the soffit up to the ridge vent. Too often dirt and debris block the vents and homeowners force insulation all the way to the roof blocking the air flow. After that is checked, follow the ducts from any bathroom exhaust fans to make sure they vent outside the home. In a best case, they go out the side of the house, well away from the soffit vents, or at least out the roof. But at a minimum they should go out near the soffit (where much of it will get sucked back into the attic) and not left venting into the attic directly. Lastly, check under the insulation to be sure there's either the paper facing of the insulation or a plastic vapor barrier between the insulation and the ceiling below.