Damp probably is limiting factor for this particular specimen, rather than shade. If you could make a raised bed to put the plant above the sogginess then you might get it to grow better or open up the site to many other candidates. 6 inches of new soil might be enough.

If the soil is damp but not boggy, climbing hydrangea might work. A shrub that might give a similar look to winter jasmine, but not as easy to train to a trellis, is an early bloomer that likes damp/wet soil- Kerria japonica, or Japanese Rose. There are single and double flowered forms.

I actually have kerria japonica in my yard and I love it.

The soil is quite boggy in the wet months. A replacement plant doesn't need to bloom at this time. Just looking for any vine/climber that likes a boggy area.

I did a bit of searching and found that clematis crispa would do but would like some other ideas too. Thanks.