The Old Testament doesn't say much about marriage.
Pretty much everything the Bible says on the subject comes from the New Testament.
I'm thinking that there's going to be a massive argument over semantics behind this statement. The Old Testament is full of laws regarding who a woman does or doesn't belong to and punishes violations as the sin of adultery. If you want to play games around the word "marriage" being used or not, it's beside the point.
At any rate, the reason I'm convinced that this belonging of a woman to a particular man begins with the promise is because otherwise, Israel's law would have essentially obligated people to be stoned to death as adulterers in what God considered a legitimate marriage: for example, if a woman ran off on her betrothed, was "taken" (or whatever word we feel like using) by another man some other place (potentially not knowing anything about the betrothal) and even consummated, and if they were caught by Israel doing this they'd be stoned wrongfully for adultery whereas God considered the "adulterous" couple legitimately married.