I think the black screen on continuing has something to do with the game not being able to load the save file properly - something similar happened to me when I was messing around with the profile files and renamed one to lowercase letters.

There is a possibility for what's gone wrong, and if it's the case then it should be fixable. What could be happening is that the HDT file connected to the profile is corrupt in some way.

Bear with me, this may be a bit long-winded - (assuming you've installed in Program Files (x86).)
First, you need to find your save files: These will either be stored in C:\Program Files (x86)\GOG.com\Theme Hospital\SAVE, OR C:\Users\(user)\AppData\Local\VirtualStore\Program Files (x86)\GOG.com\Theme Hospital\SAVE
"(user)" is your computer username in that path.

In this folder there should be a file called "HISCORE.DAT" (leave this one alone) and then a few files with your profile name: NAME.HDT, NAME.H99 and NAME.H# (where # is a number from 1 to 8 and NAME is the nickname you use to start the game)
Copy the ones containing numbers somewhere else to back them up, and then delete the originals and the HDT file from the SAVE folder.
Then, start a new game with the old profile name, and save it in slot 1. Quit the game, and the SAVE folder should have new entries. Delete NAME.H1 and NAME.H99, so that the SAVE folder contains only HISCORE.DAT and NAME.HDT. Then, move the backed-up files back to the save folder.
When you next start the game it would probably be wise to load from one of the numbered saves rather than the "continue" button. Hopefully then it will all be back.

Hope I've not been too confusing with this, and here's hoping it will work!