The new sign in page is apparently causing problems for some folks; most of them seem to be using Microsoft Edge or Explorer browsers. The "keep me signed in" box on the sign-in page does not seem to function correctly.
There is another "keep me signed in on this computer" checkbox in your Account: Site Preferences: Other General Preferences. Check that box and hit "Apply". It seems to work for some people, at least until they sign out again.
Here is a direct link to that page, it's all the way at the bottom:
https://my.ebay.com/ws/eBayISAPI.dll?MyeBay&CurrentPage=MyeBayPreferences
If the box is already checked, uncheck it, "Apply", sign out, sign back in, and recheck it, then hit "Apply" again.
