could they be too wet and maybe too cold potting on is a bit traumatic for them

Pamg is right....compost too wet....need to keep them on the dry side until they have established and grown a bit..

I agree, it is important when transplanting to only water around the edges of the pot/container and avoid watering too close to the stem.This encourages the root system to look for the water and promote healthier and stronger roots. I've got some mail order begonia plugs arriving soon and I shall be planting them straight into final position to avoid transplanting again.