Are there special supplications for facilitating marriage to a specific person after trying several previous supplications without success, and what should be done about the feelings of sadness and depression resulting from the engagement not being completed?

There is no specific supplication for marrying a particular person. Supplication in general is beneficial for every legitimate request. One must believe that marriage and other life matters occur by Allah's decree. Allah is more merciful to us and more knowledgeable of what is best for us. Entrust your affair to Allah and have a good opinion of Him, for He will prepare what is good for you.