Thank you, Linc! Your tip helped me to resolve (or at least work around) this issue, and I'm sharing my experience in case it helps anyone else.
tl;dr: Use the third-party Pref Setter app to change the "Disabled" flag from "true" to "false" for the Screen Sharing and File Sharing .plist files.
The command line you suggested returned the following result for me (OS X 10.12.5 Sierra):
disabled services = { "com.apple.screensharing" => false "com.apple.smbd" => false}
From my regular (non-Admin) login, I then used EasyFind to search for the associated .plist files (com.apple.screensharing.plist and com.apple.smbd.plist), and found .plist files in "\Library\...," "\System\Library\...," and "\Username\Library\... " When I did a "Get Info" on these files, neither the System (root) user nor the Administrators group was allowed Write access. I granted this access, restarted the Mac, and then checked the File and Screen Sharing checkboxes in the Sharing Preferences, but upon restart the boxes were unchecked.
I then deleted the .plist files, believing that they would be recreated on startup, but upon a restart and opening System Preferences->Sharing, the File Sharing and Screen Sharing checkboxes would not allow me to check them. So, I replaced these .plist files from a backup and restarted again.
One by one, I opened each of the replacement .plist files with Pref Setter. When I did, I saw that a "Disabled" flag was set to "True" for all but one of the .plist files. I set the flag to "False" and saved the edited file. I had to save to my Desktop as I lacked the appropriate permission to save directly to the source folder. I renamed each original .plist file to ".plistOLD," then dragged the edited .plist to its appropriate folder. Then, I used the latest beta of BatChmod (1.7 beta 5 at this writing) to change the ownership of each edited .plist to "root," gave "root" and the "Administrators" group Read/Write permission, and the Everyone group Read (only) permission. After a restart, the File Sharing and Screen Sharing checkboxes remained checked.
Admittedly, this was a brute-force approach. I could have edited the "Disabled" flags for each .plist one-by-one and restarted each time to see the results, rather than editing them all at once. I also haven't tried unchecking the File and Screen Sharing checkboxes and restarting. I simply ran out of patience and time, and for now, I'm happy to have File and Screen Sharing settings remain "on" across restarts. In the unlikely (for my needs) event that I need to change these settings in the future, I now know how to edit the .plists. It's an ungainly approach but an improvement.
Again, thank you, Linc, for providing the key to resolving this. I had been unable to find the corresponding .plists before your tip.