I am not quite sure what a giant dogwood is.
My experience of selling dogwoods and their care on our sales beds is that if to wet they will loose root and potentially die. If grafted and you are getting suckers potentially the top is dieing- usually due to a bacterial or fungal problem. The plants energy to keep alive travels just under the plants outer skin / bark. If the flow is then interupted by such an illness the plant will die above the decayed area.
If the suckers themselves are growing well and do not look like the top growth-ie not curling or drooping then clearly it is not a root issue however remember plants do not always wilt because they are dry- they also wilt when to wet as they loose their root hairs.
If the new shoots are suckers as suggested you must remove them as they will simply take the available energy away from the top you are trying to save.
Hope that helps

Thanks for your reply, the dogwood -Cornus Controversa Variegata-is grafted. The "suckers" look completely different and are growing very well. Should I remove them now or leave them until the autumn. Someone told me they support the root system.

Ah the wedding cake tree !!! A lovely dogwood if you can get a good shaped one.
Many plants are grafted onto rootstocks for a number of reasons. Often to reduce vigour or increase vigour- especially variagated plants which are often weaker in habbit.
The growth you see if totally different from your required plant is clearly suckers from the rootstock. Your not growing the rootstock out of choice. Its growing on its own and will sap any strength your cornus should be sending to its top growth. You should remove the suckers straight away but also investigate why your top growth is failing. If the suckers are growing well while the top growth droops then clearly its a problem with the plants stem- maybe the graft.