How soon before you move in will the house be painted? Paint fumes are not near as strong as they used to be and the birds should be able to safely move in a couple days after the painting has been done. If the house has been closed up and you can still smell the paint then open some windows and wait a day to move them in.

As far as the egg laying- first make sure she has a cuttle bone and she's using it. If she has one but isn't using it move it to where she is laying her eggs, she may use it more there. Secondly if she's in the same cage with the male I'd suggest separating them right away. He could very well be the reason she is laying so many eggs. As others have said taking the eggs out right away will only encourage more egg laying. Also the longer days are encouraging her to lay the eggs too. I don't believe in covering a cockatiels cage because they are so prone to night fright (which often occurs when the bird can hear things outside her cage but cannot see them because of the cage being covered). You can try keeping her in a dark quiet room for a few extra hours each day- try to make it a room she is familiar with though as a new unfamiliar room may also cause her to have night fright (something you'll want to watch for in your new house, especially for the first few nights).