Moving means you have to prioritize the things that you want to bring along with you and set aside the things that you can live without. Remember that you goal is to stick to your budget without leaving behind the things that you want to bring in your new place.
For instance, packing should be done a week or so before the move. This will give you a clear idea of the things to bring and how much will it actually cost you. While there are estimates over the phone. It would be best to ask your movers to come in your place for a more detailed estimate.
Just because you are packing ahead do not always mean you have to seal it. If there are things that you may be using daily, put them in a separate box that way you get to have easy access to it should the need arise.