If you are getting verification checks every time you sign in, that might be a sign your browser is not saving or keeping cookies properly, or that your IP-address is changing. Check to make sure you are not erasing cookies when you close your browser. If not, and the problem persists, it might be worth trying to manually erase just your eBay cookies once.
Some browsers are better than others when it comes to this sort of thing. Try another browser, or a private or incognito browser window. Or try using the eBay app.
You may be able to get out of a verification or sign in loop by signing in using a social media account if you have previously set up such a thing.
eBay does not have a number you can call or an email address for these kinds of issues, but you can request a chat with an agent or a call back from customer service. I am not sure if customer service is in a position to provide much in the way of tech support for new issues.
You are probably better off posting here regarding such matters to get advice from other users that have run into the same sorts of problems rather than front line customer service workers who are reading from a prepared script.
