I used to offer the following songs when I sang at weddings:

The Rose (not at all sad but assumed by many to be)
There'll Never be Anyone Else but You for Me - (Ricky Nelson song from the 60s.)
Love is a Golden Ring
Have I Told you Lately that I Love You.
Dream.
Devoted to You.
Remember Me I'm the One Who Loves You.
Always
My Love is Like a Red, Red Rose
Wild Mountain Thyme (with a few word changes in the verse about finding another if it doesn't work out.)

All a bit dated but they still work for the right people. Cheers, Joy