If the stairs are many, as in a staircase, I would re-carpet them, even if you choose to use laminate on the floors. The two will blend together.
Laminate would be labor intensive on a staircase, not necessarily cheaper than carpet to purchase and would bounce the sound around the walls going up the stairwell- something most people won't be drawn to.
If your concern is keeping new carpet looking good while you are packing, showing the house and moving, here are two suggestions:
There is a company called Protective Products (online) that sells rolls of what looks a bit like saran wrap. It is temporary carpet protector with a light-tack adhesive that will stick to but not harm the carpeting.
Up North, it is commonly put down in most model homes, along the traffic pattern of the carpet.
I have used this product in two different homes I have sold. It can be safely kept down for about 6 months with no harm to the carpet. I have used it on staircases and it is not slippery.
It is helpful to have two people working together when laying it down, and well worth the effort. Potential buyers were impressed that this care was taken.
I would also wait to install the new carpeting until anything else that needs to be done is completed. The new carpet would be the finishing touch before showing.
My other suggestion would be to offer (as part of the sales contract) an amount of money toward replacement carpeting of the stairwell.
If you should buy a roll of this stuff, you may very well have some left over to place down in your new TV home, before you move in. We did, and it helped a lot until we got settled.
Good luck on your new adventure!
