You can't keep cookies from a Private Browsing mode session. If you leave Private Browsing mode or exit Firefox while in Private Browsing mode then all cookies are removed and the cookies that you had in the previous regular session will be reloaded (assuming that you didn't close Firefox).

Thx cor-el for the response to the cookies question, my follow-up question then is: What is the best way to keep selected cookies and remove the rest?

You can protect specific cookies with the CookieCuller extension. The cookies are only protected within that extension, so other means (Clear Recent History) will still remove them. You can let all cookies expire if you close Firefox and keep the protected cookies. If you use the build-in cookie manager then you can only make an allow exception for a specific domain, but with CookieCuller you can protect specific cookies.

• CookieCuller: https://addons.mozilla.org/firefox/addon/82