The only other thing you could do, besides FF2's bookmark suggestion, is to write down the location number of the page you last read and then when you open the book you would need to use the Menu "Go to" option and input the correct location. Perhaps not quite as complicated as using bookmarks and at least you wouldn't need to remember to delete your old bookmarks.
Kudos to you and your partner for being willing to share a Kindle
and to share the same book at the same time. I've never shared the same paperback book with someone (as in, reading it at the same time,) let alone the same Kindle eBook on the same device. Guess I'm just too selfish to take sharing that far - you can read the book
after I've finished it is the way I would be dealing with this.
Holly