Is the fan safe to operate?
A burning smell from an electrical appliance (other than a toaster) is a sign that it is time to replace the item.
Sizzling sounds combined with burning smells are sometimes a precursor to burning the house down.
Melted electrical insulation reduces the electrical safety of the item and could lead to electric shocks (and possibly death).
I'd stop using it until it is removed (with power off at circuit breaker) and repaired or replaced.
What should I check for?
I'm not familiar with these units but, from your description, it is probably a fault in a switch operated by the chain. It may be that dirty contacts at one position are causing arcing. Arcing causes a sizzling sound, Arcing causes heating and that causes barning smells as nearby non-metal parts are burned (eventually they may melt or catch fire)
Arcing is when sparks are jumping across a small air gap where the switch contacts are pitted, corroded or dirty.
In principle it may be possible to clean-up the contacts (which may only be a temporary cure) or replace the switch (if spare parts are available). You might need to replace any other parts damaged by heat.