Unlike the other answers, I am not familiar with the different parts of the ceremony itself, but I would recommend the following songs in general for mushy feelings -
David Bowie - "Be My Wife"
Sparks - "Marry Me"
Magnetic Fields - "The Book of Love" (I personally would play "Take Ecstacy with Me")
Phyllis Dillon - "Don't Stay Away" (it is about unrequited love, but it is s beautiful song and the sentiment is sweet)
Betty Wright, "Just You"
Sugar Pie Desanto - "I Love You So Much"
Willie Nelson, "Can I Sleep In Your Arms"
Little Willie John, "Need Your Love So Bad"
Lefty Frizzell, "I Love You a Thousand Ways"
Solomon Burke, "I'm Hanging Up My Heart for You" (he will stop his slipping around for you)
Dusty Springfield, "Make It With You"