If your booklet is to be printed by a commercial printer supply them with a PDF - they will do the imposition.
Supply it as separate pages (not spreads) and if you haven't been given a spec by the printer use the preset PDF/X-4 and tick Use document bleed settings.
You might find 24 pages is cheaper to print than 20 pages!
When you shift pages the master page will stay with the page you are moving, but you have to be very careful where your cursor is when you release the mouse. It might be easier if you increase the page thumbnail size in the Pages panel: Go to the flyout menu in the upper right > Panel Options.
I am actually planning to print a batch on my own printer, which is why I need to do the alternate layout..
Barbara, where would the cursor have to be that would make the difference between the page keeping the master and losing it?
Best to print from PDFs, rather than from InDesign. You can export to (in this case) High Quality Print from the InDesign presets and arrange into printer spreads in Acrobat Pro.
Sorry for the delay. Your cursor should be on top of the page you where you want to move the page. For example, when you're dragging page 19 to page 3, your cursor should be on top of and kind of in the middle of page 3. You should see a little arrow that looks kind of like this |-> with the arrow pointing to the right (I assume that page 19 and page 3 are both right-hand pages). I've found that when moving pages down instead of up, the arrow |-> sometimes doesn't appear but it still works.
However, now that you've explained that you are doing this to set up pages for printing, why not try using the File > Print Booklet command.