Some thoughts based on what you've said so far:

1) This bike may have a thread-on freewheel instead of a cassette. If the wheels are original, it's likely a freewheel.
2) If you have a freewheel and the chain slips on the smaller rear cogs under load, then the smaller cogs and/or the chain may be excessively worn. If that is the case, consider sending the freewheel to Pastor Bob (Freewheel Spa in NH), or try to locate a gently-used or NOS vintage replacement from Suntour, Sachs-Maillard or Shimano. If you have a cassette then replacement of one or multiple cogs will be the solution. Normal caveats apply (buy once, cry once).
3) Have you tried to activate the derailleurs (e.g. get them to move) while the bike is stationary or without wheels? What does the shifting feel like then?
4) Have the springs in the derailleurs been lubricated at all?
5) Definitely get rid of any slack in the cables. It's bad for friction shifting, too.

Peugeot Carbolite 103 is a step up from the high-tensile and mild steel used for older versions of that bike. It should have good riding properties if the geometry is slack.

That frame is likely too big for you, unless you're over six feet tall.