OP: if that's also the original freewheel for the bike, you may have backed yourself into a bit of a corner.

If/when a bike's chain and sprockets/chainrings wear together long enough, they can become effectively a "mated for life" pair - and neither will work properly with new counterparts. They then have to be replaced in pairs.

This is particularly true if you ride primarily in one or two rear gears and on one front chainring. In that case, if you do so long enough the chainring can also become worn enough to be mated to that chain. (Years ago when I was much more naive re: cycling, I found all of this out the hard way.)

Sounds to me like that's what's happened to your bike. If so, and if keeping a 32T rear cog is of overriding importance to you, as I see it you have four options - only one of which is fast, cheap, and easy. And, unfortunately, that option is a "kick the can down the road" option.

Option 1: assuming you still have the old chain, use a 6/7/8-speed quick link to replace the broken link in the old chain, and keep using that chain. This is the "kick the can" option, but might work for a while - but may also accelerate wear on the front chainring(s) and rear cogs. It's also only feasible if the broken link was an outer link.

Option 2: find a good used or new 5-speed freewheel (eBay, co-op, elsewhere) with a 32T largest cog and pair that with a new chain. Unfortunately, used will very likely be easier to find than new with a 32T largest rear in 5-speed. That may take a while.

Option 3: re-space the bike's rear dropouts to 126mm, rework the rear hub as required (may need new axle and some spacers), re-dish the rear wheel, and get a new 6- or 7-speed freewheel and chain. Fair amount of work (and $$$ if done by a shop), but might be worth it to you. It should also let you keep using the bike for a reasonably long time.

Option 4: have the old freewheel overhauled (some cogs will likely need to be replaced due to wear) and use it with a new chain. That's unfortunately a dying skill, but BF member pastorbobnlnh still does that. His reputation for such work is stellar. Dunno if he does Shimano freewheels or not; you can contact him via PM to ask if he doesn't see this thread. His website is

https://www.freewheelspa.com/

Hopefully it's only some of the freewheel cogs and not also one of your chainrings that's worn enough to slip. In my case, I ended up replacing both.

Best of luck.