AB2012: First thing is how are you using DOSBox:-
1. If you have it installed manually (you installed it from DOSBox.com and are adding games manually to it), then the DOSBox config file is usually at C:\Users\USERNAME\AppData\Local\DOSBox\dosbox-0.74-3.conf so open that in Notepad.
2. If you are just using it via how GOG packages it, then usually they'll have the config files in your game folder ending in .conf which are DOSBox configuration files specific to each game. The main one is usually called "single.conf" or something like that (the name differs per game).
Either way, if you open the .conf files in Notepad, what do you see in your under mixer and sblaster categories? Eg, under mixer, make sure that
nosound=false and that the sample rate is something like 44100. Under sblaster category, make sure than
sbmixer=true, and that defaults are something standard like sbtype=sb16, sbbase=220, irq=7, dma=1, hdma=5, oplmode=auto, oplemu=default and oplrate=44100.
The fact that it's happening in multiple games suggests that there's something overriding it beyond one individual game's config file (possibly a Windows 10 issue blocking DOSBox from accessing hardware)? And the fact it's working in other games rules out hardware fault. First thing I'd do there is backup any save game files, uninstall a problem game (and select "No don't keep my saves" so GOG deletes existing config files), then reinstall it and see if that works. Or if you installed DOSBox manually, go to the DOSBox install folder (usually C:\Program Files (x86)\DOSBox) and run Reset Options.bat. If all else fails, contact GOG support.
In the meantime, try the excellent
GZDoom source-port for Doom 1-2 (install it then drop your DOOM / DOOM2 / PLUTONIA / TNT WAD files into it). You should at least get audio back in Doom as well as a more pleasant gaming experience with mouse-look, 35fps cap removal, etc.

TheCoop: All of that was checked on. The installs of DOSBox were using the same settings, so it wasn't a conflict in that way. And Windows (as I learned) wasn't blocking anything.
TheCoop: I found that early this morning, but that post didn't explain anything about where they did the fix (in DOSBox? in Windows command prompt?). However...
After doing A LOT of google searching once I got up, I found one post on a forum buried in a lot of text that mentioned where this SDL_AUDIODRIVER environmental variable could be found. My Google-Fu isn't the best in the world, so it took a good while. But, it's located in your computer's registry. Specifically for Windows 10...
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Session Manager\Environment
I had to change the key there from "directsound" to "dsound" and then POOF!... sound effects in DOSBox. The Immortal music played, the selection sounds in Ultimate Doom played, and all was right in the world. I have no idea why that key was changed or when (a Windows update maybe?), but it was. Gotta love it when something so small gets altered behind your back and creates a sizeable problem.
So hopefully this post will help anyone else who's not as savvy with this kind of thing. POst after post mentioned the SDL_AUDIODRIVER thing, but 95% of them didn't give the most important bit...
where the hell it was lol.
That said, thanks for trying to help, guys. It was much appreciated :)
I would sincerely recommend re-creating this post in the Doom sub-forum as well. This may save someone in the future a lot of grief and headaches trying to solve it.