Kiddushin 2:10-3:1

Kiddushin 2:10

If a man betroths a woman using terumah, tithes, the portions given to kohanim, or the water or ashes of the red heifer, she is betrothed. This is true even if a Yisroel does it (as opposed to a kohein or a Levi).

Kiddushin 3:1

Let’s say that one man appoints another as his agent to betroth a certain woman for him but the agent betrothed her for himself. In such a case, the woman is betrothed to the agent. Similarly, if a man says to a woman, "You are betrothed to me after 30 days" and another man betrothed her in the interim, she is betrothed to the second man; if she is the daughter of a Yisroel betrothed to a kohein, she may eat terumah. If he said, “You are betrothed to me from now and after 30 days” and another man betrothed her in the interim, she is quasi-betrothed (i.e., her status is in doubt). In such a case, whether she is the daughter of an Yisroel betrothed to a kohein or the daughter of a kohein betrothed to a Yisroel, she may not eat terumah.