Okay, I admit, I'm normally fairly good with old DOS games, since I always enjoyed going back to play them, but this one had me incredibly stumped. I finally stumbled over the "fix" to the problems between the Dragonsphere CD version and DosBox, and thought I'd share it here for others. (Credit for most of this goes to the various threads and suggestions I found while browsing this forum)

Step 1: Own a copy of the Dragonsphere CDROM. =)
Step 2: Assuming that your hard-drive is C: and your CDROM drive is D:, create the following directory C:\DRAGON
Step 3: Copy the ENTIRE directories of D:\SPHERE and D:\RESOURCE into your C:\DRAGON folder.
Step 4: Download and extract the original INSTALL.EXE from the Dragonsphere floppy version into your C:\DRAGON folder.
(http://www.houstondragon.com/dragonpatch.zip or see attached file)
Step 5: Run DosBox, mount your C: and D: drives, and run the INSTALL.EXE program. Setup your CDROM, sound, and speech configuration and exit the installer.
Step 6: Run DRAGON.BAT and enjoy the game!

This has been tested with the "Player's Choice" version of Dragonsphere, and under DosBox 0.65. There have been no other problems, so far.