We are running a ukulele group. True-love is a competent teacher so we have an advantage. We try to help the group understand the limitations and the advantages of using ukes. We began with simple songs -- one chord to two chord to three chord and so on. Thing is -- it's easy to play simple songs with few chord changes but it quickly becomes boring. Finding songs with interesting progressions is the key. I agree that most sea songs don't work very well. Short-haul shanties are the best bet if you really are sold on using sea songs. Calypso songs sound good if you work on the beat. We're having fun with jug-band music and with some blues. Also some 60s rock.
Proud Mary is really easy with a good beat and it sounds good even with beginners.
Good luck Sarah. If you'd like a list of our workable songs let me know.
Cheers, Joy