Do they mean tidy up as in clutter or tidy up as in marks etc on the walls?
If you haven't got the same colour paint, magnolia the room - check to see if you have a scrapstore/paint recycling centre near you as you can pick up paint very cheaply there.
If you need everything in your home, I would honestly go for the storage unit rental option. Box up stuff you are not using on a daily basis and store it. Before you move into your new home, go through everything in storage and be brutal - it really helped our family of 6 when we moved house and wanted to get rid of lots of junk.
Get the kids, if they are old enough, to declutter any broken toys, damaged books, things they are too old/no longer want. Anything good enough to sell either list on FB or Ebay, or if you don't want the hassle, drop into a charity shop. even one bag out every time you go shopping is better than nothing.
People aren't going to be bothered about things stored in a garage or loft - chances are they won't look in the latter when viewing. But they do want to imagine themselves in the house. Although you have a large family and I know this puts additional pressure on available space, people won't necessarily think 'a family of 6 lives here and we are only 2/3/4'. They will see a home with potentially little space for growth.