First, in Document Preferences (in Word, the gear button), for troubleshooting, make sure you're using Fields and not Bookmarks. Now, try pressing Alt+F9 (press it again to undo).

You should notice that the new citations you insert will turn into large blocks of code. The old citations should do the same thing, but I suspect that they will not change. If that is the case, then at some point you probably used the "Remove Codes" button, which turns live citations into plain text. Alternatively, you may have saved the document in a format that does not support field codes (e.g. RTF). Unfortunately, neither of these actions is reversible, so the only way to retrieve the old citations is to find an older backup of the document that still has field codes. Otherwise, you would have to re-insert the citations.

This certainly _should_ work. If you were getting errors about citations no longer being present in your library, then this would indicate a problem with the way you synced/transferred Zotero. Since you're saying that the citations are plain text (I assume you're concluding this by positioning your cursor on a citation and clicking Edit Citation, which would then tell you "You must place the cursor in a Zotero citation to edit it"), then something happened to your document. If it is still working on the old computer, then I would just copy the document again from the other computer. You can also see how the current document behaves on the old computer.