I bought Disco Elysium when it came out. I was only able to download the offline files to play the game. Fast forward to Cyberpunk. I bought that game as well. Yet when I go to play either game on GOG I hit the play button then it disappears and turns into the buy button. GOG shows I have purchased the games. Disco Elysium even has how many hours I have played the game, yet I cannot play these games. I have emailed GOG multiple times and have got no response every time. Can someone help me? Am I screwed out of these games?

I assume both games appear here? https://www.gog.com/account ?
If yes you just can download the offline installers by selecting the game , clicking the arrow down --> View Downloads --> Click Arrow next to Download Offline Backup Game Installers. Then just download the offline installer files there and start the game from the game.exe in the Games Folder.


Regarding clicking the "play" Button i'm guessing you are talking about starting the game from GOG Galaxy?
If yes are you sure you are logged on into GOG Galaxy with the same account as the account which owns the games?
Also if the games are already Downloaded and refusing to start with the Play button from Galaxy just select Manage Installation --> View Game Folder under the games options in Galaxy and see if you can start the game from the folder.
If this does not work try the "Verify/Repair" button

When I go to manage installation the only options that come up are Configure and uninstall (which is grayed out for some reason). The games show up in my account. They are also installed on my computer. They show up as installed in GOG galaxy, yet when I hit play the button turns into the buy button. On GOG is shows that I have the games in my collection but I still have the option to purchase them. If i push the buy button on GOG galaxy then the options under manage installation appear but all of them are grayed out.

Your profile shows 0 hours played, so clearly something's off. Note that GOG Galaxy has multiple executables spread across the Program Files (x86) and ProgramData folders. You should make sure they all have access through your firewall (this may be excessive, but there's no knowledgebase article that details which executables need access and support was unreliable). Additionally, the game executable itself needs access through the firewall for things like achievements to work. Try a fresh reinstall of GOG Galaxy:
1. Uninstall from the Windows Control Panel.
2. In File Explorer, paste C:\ProgramData\ into the address bar. If there's a GOG.com folder there, delete it.
3. Now paste %LocalAppData% into the address bar. If there's a GOG.com folder there, delete it.
4. See this JSON file for the link to the latest offline installers. The second one is for Windows; get that and install it. After opening a support ticket, you should receive one or several automated replies. If they're not in your inbox, check the junk folder. If there's nothing there either, change your e-mail address and open a new ticket.
Once you've received an automated reply, be sure to read it. One or more of those requires a reply from you to keep the ticket open.
In those e-mails there should also be a link to the support ticket on the site. Be sure to bookmark that since there's no longer a way to see a list of all your support tickets. With that link you can see the status of the ticket, conversation history, and you can reply or add more information (this is the same as replying by e-mail).

The only emails I have gotten are advertisements and spam. I have checked all email fplders.

Dumb question time. The account you're logged into Galaxy with is the same as the one you purchased the games with, yes? Double check that.