I am very much in the same boat as OP, so I have been reading the answers closely,
I have tried to tread the path between nagging and positive encouragement, and to a certain extent it is working.
I too suspect that OP's husband is scared and not liking this reminder that none of us live for ever.
Mine has admitted to feeling fairly useless and guilty about leaving so much to me, so I have already tried to suggest things I know he can do, and to ask for help with things I could manage alone, but which are within his capacity.
OP can you ring and make an appointment with your husband's GP about the dizzy spells? They can be caused by a variety of different, treatable and not dreadfully serious complaints, but until you know what causes them, you and he will worry about them.
Sounds to me as if your DH like mine deals with worries by ignoring them and watching mind-numbing TV instead of getting off his backside and dealing with the worry!