Hi Sheila,
I've just been speaking with my wife about how we brought our things over. It came in two batches:
When we first came over we brought our crockery, cutlery and other essential items over in a shared container (technical term 'Groupage'). We used these in the house we rented while ours was being built.
We brought the rest over in a 20 foot container when our new house was nearly ready - furniture, kitchen hob, microwave & oven, etc. (We bought the hob etc. in the UK because they were a lot cheaper there than they were in Cyprus - I don't know if it's still the case).
We didn't have any problems with damp - all our things were wrapped or boxed and then wrapped in polythene and strapped on pallets.
But I do remember going to the warehouse at the Limassol port to identify our belongings. Some people had just dumped their things in boxes without bothering to sort them out and I recall seeing a hosepipe that was leaking water.
But if you use a decent shipping company, everything should be OK.
There's a company that advertises in the Cyprus Weekly - 'Britmovers'. Bob was Over's agent in Cyprus - it might be worth giving him a call.
Cheers,