In Device Manager do you see a device something along the lines of "ESS Allegro DOS Emulation" this would be "soundcard" dos games would use so to speak from within windows.
I don't own the card so not sure if this only included in the VxD drivers and you would be using WDM drivers been on WinME

The real mode patch would probably be easiest option to get the card working though.

I installed WinME with a ESS 1868f. The drivers came with WinME, and they were WDM. I found that the "Win95" drivers on vogons drivers site were VxD. You cannot go into the device manager, and force "update driver" and select that driver because it will tell you that the newest already is installed, like you can do on Win98. So I moved the INF file out causing this of the \windows\system directory and deleted the driver in device manager and then ran add/remove hardware, and then I could use the VxD driver, which work with DOS programs correctly.

I do not know what drivers are available for your PCI card.

It should not require any DOS driver at all if you were staying with WinME, I presume. Most legacy PCI sound cards with SB emulation could just provide the same emulation through VxD for VDM sessions. WDM audio class driver simply offers more flexibility so that legacy-free audio hardware (for eg. AC97 and HD Audio) can support SB emulation through OS without spending the silicon on SB emulation logics.

You can try load device drivers from WIN.INI instead of config.sys
missing in WinME.

see below for example adding to WIN.INI

[windows]
load=
run=
NullPort=None
device=HP LaserJet 4050 Series PCL,hpbffb1
device=C:\WINDOWS\TWAIN\MSPNRI.SYS
;
;you can add ESSAUDIO.SYS here
;
device=C:\ESSAUDIO.SYS

[Desktop]
Wallpaper=C:\PROGRA~1\PLUS!\THEMES\1701D22.BMP
TileWallpaper=0
WallpaperStyle=0
Pattern=(none)