Should be okay as long as the rhododendrons are big enough to not look like they are being inundated. Avoid chopping chunks out of their roots to plant it. Maybe start it nearby and let it spread among them, from there.

Once you plant Sweet Woodruff, you will have it forever, whether you want it or not.

I think using it as a groundcover between rhododendrons would be fine, but it would be a good idea to keep it off the rhodendron roots themselves.

As far as keeping the soil moist, all ground covers shade the soil somewhat but they take up water as well. The net effect is they make the soil drier, not more moist.

I have also never seen sweet woodruff climb anything and having any plant grow up the trunks of rhodendrons is not a good idea - here it would tend to make things overly moist, an invitation to fungus, etc.

Rhododendrons and sweet woodruff both have shallow roots. They can coexist but not an ideal match. Sweet woodruff is hard to control and will spread. If you plant it, plant it between the plants not under them. Use a mulch under the rhododendrons. Sweet Woodruff is somewhat deer resistant.

Astilbe, Hepaticas and Cyclamen hederifolium do well underneath rhodies. Hardy
geraniums also do well.

Thanks for the advice, I decided to move the SW to another location so as not to incur any problems in the future.