Ebay shows I signed out but it will open up w/o having to sign in the next time we use it.

[sauce4050]

I can sign out and it shows we signed out but in clicking on the eBay logo it opens right open again. 

[eburtonlab]

You should be able to sign out of your account in your browser by erasing your eBay cookies.

If you are using a public machine and are concerned about signing out, try using a private or incognito browser window for eBay; when done, closing the window will end your session and leave no cookies or history behind.

[papermoon.lady]

I did that once (trying to use an incognito browser) and they thought someone was trying to access my account and locked my account. They sent a code to my email address and it was OK (It's really you, - my first name) when I used it to unlock my account, but sometimes it might cause problems. I had used an incognito browser before, so it does not always happen.

Deleting cache and cookies and restarting the browser should definitely work though.