By coincidence I was doing this last night because I got a new eReader (non-Kindle). I found the most reliable method was to go the the "Manage Content and Devices" page on my Amazon account and manually download each book. This method downloads a single file for each book to your machine and its in a format Calibre and DeDRM can work with. It took a while even though I only had ~80 books but it was worth the effort.
That page should have a list of books you own. On the left side of each row for each book, there should be a light gray button that looks like: [...] It's under the "Actions" column, second to the left. Click that and there should be a popup that contains a link "Download & transfer via USB"
I believe this link will only be available if your account has a real kindle (e.g. eink or fire) registered to it. It encrypts the downloaded file for whichever registered kindle you select from a list of all your registered kindles and apps, with the apps grayed out so you can only select actual kindles.
