Honestly the best way to run Chex Quest is through a Doom Source Port. Because Chex Quest is really just a modified version of Doom it works just as good as normal, if not better. This should also fix your sound issue.
Admittedly, most if not all Source Ports have made vast improvements over the original Doom engine, and thus provide a slightly different environment, so if you're looking to play the game EXACTLY how you used to, then it seems you've got things worked out.
However, there is a reason for all the improvements made on the Doom engine and they make playing any game compatible with the engine (Doom 1 and 2, Heretic, Hexen, Strife, and of course Chex Quest 1 and 2) a better experience.
There are many Doom source ports available.
I'm only going to link to the ones I've used personally, but a Google search will turn up many more. Shop around and choose one you like best.
Anyway here they are:
http://zdoom.org/
http://legacy.newdoom.com/
If you need any help getting Chex Quest either to run both should have plenty of support on their respective websites.
If you can't find what you're looking for on their official sites/forums then email or PM me and I'll help you further.