@Greykyne There is a disconnect somewhere. That error typically means one of the following:
You either don't own the content (another EA account on that console does)
The content is not downloaded correctly (you can always uninstall just the content from manage games and apps)
It was part of EA Play and you no longer own that.
With Dragon Age and Mass Effect, the image you attached generally indicates an issue with the DLC, your account, or a save file you have has content you don't own. Regardless of your account owning it, that is when the errors happen.
I launched the game on the Series X and had no issues which means it isn't a game issue as I have tested both a random personal act and a work act.
When you launch the game and go into the options, if your EA account is logging in correctly, you will see your EA account in the "account" section of the settings.
If you are not, the game does not recognize the account. That is the part you have to figure out. Since the 360 servers are closing down, it doesn't recognize it as a 360 game any longer.
When you launch the game, can you press start and select and bring up the 360 dashboard? It should show the account you are playing on. If not it isn't connecting. If it does, and you go back into the game settings, under the account, and it is not logged in, there is a disconnection in the EA account connecting to the game.