Clearing excess wax from chain links
Typically when hot paraffin waxing I dip the chain for 15 min, then remove and hang. I typically hang the chain very very gentle as I don’t want to dislodge the liquefied wax from the rollers/bushing interface. I don’t wait for the pot of wax to cool. When hanging the excess wax immediately drips off, and rarely do I have  more than a thin coating between the link plates. Only the last link has excessive wax which is quick to clean out manually, once the chain has cooled back to room temperature.  

While excess wax does drains off, enough remains in the bushings and rollers as I typically get hundreds of KM before the chain makes the painful “no lube” sound. Chain longevity has also been good using this approach, about twice the mileage I get from running the same brand/model of chain with chain lubes.

About the only issue I find is breaking the frozen links after the chain has cooled as the solidified wax sticks the inner/outer link plates together. To get the links pivoting again I manually run the chain over a small diameter metal shaft. This forces the links to bend and quickly gets the links free so that the chain runs properly when I put it on the bike.

I would suggest the way you let the chain cool off, after being waxed, is contributing to the links becoming filled with wax.