Can you become engaged while waiting for an annulment?
Moral law would indicate that, apart from exceptional circumstances, the answer is no.
A declaration of nullity—commonly called an annulment—establishes that one is free to marry. Since marriage enjoys the favor of the law, until the annulment is granted, the presumption is that a person will not be able to marry. He should act in accordance with that presumption until the contrary is established.
Failure to do so can lead to dire consequences. If you get engaged prior to receiving a declaration of nullity and it turns out that one is not granted, then you will be in a very difficult situation. You will have gotten your hopes up for a situation that cannot come to pass. You will have done the same thing to another person—your fiancée or fiancé—whom you profess to love. Real love would not put another individual in such a position. Finally, many Catholics in your situation would find themselves in a near occasion of mortal sin, as they would be tempted to “jump ship” and have a wedding outside the Church. This, of course, would result in an invalid union and a state of living in grave sin.
All told, it is far more responsible to act in accordance with the presumption that you are not free to marry until it is shown that you are. It spares everyone a world of hurt and temptation.