Check the condition of all bearings (BB, headset, hubs) on the Schwinn. If you need all four rebuilt or replaced, that may affect your decision. (The rear axle is probably bent and you probably need a new chain and freewheel.) Check condition of wheels. Steel rims and galvanized spokes may not be serviceable.
See if you can convert to 700c wheels (brakes might not allow that). Newer wheels, especially with a freehub, will make the bike more tour-worthy. BB and headset can usually be replaced or rebuilt easily enough if you have the tools.
The Continental was my first touring bike and I enjoy seeing them rebuilt and back on the road, but many of them just aren't worth working on any more.
I wouldn't consider touring on a MTB with flat bars, and certainly not a suspension fork. Otherwise, the geometry and components would probably be fine.