If you can discern a length above one inch for an individual link then your chain is almost definitely extremely worn... the usual measuring strategy is to measure 12 links centre-of-pin to centre-of-pin, and if it is longer than 12-1/16th inch, the chain should be replaced.

However, if the chain is excessively worn, then the cogas and possibly the chainrings are also worn to match, and a new chain will not mech properly in the old gear teeth... on modern 'hyperdrive' ramped and pinned shaped cassette teeth, a new chain on a worn cassette will skip under pedaling forces. Perhaps an older non-hyperglide freewheel will hold the chain better, but probably still not work too well. You will likely need a new chain and freewheel, and possibly new chainrings.

Whether or not this solves your shifting dilema is unclear - older bikes without indexed shifting or ramped and pinned chainrings were not so smooth shifting, and learning to shift correclty each time without missing a shift or overshifting was one of the skills you used to need to consider yourslef a competant cyclist back in 'the day.'