I could see how juggling six accounts would lead to verification fatigue quickly.
Depending on how you use those accounts, it might be possible to minimize signing out and back in. If you have one account you use the most, perhaps you can leave that account signed in using your primary browser. Perhaps have a secondary browser for your next-most-used account. If you have accounts that you only have to check once in a while, perhaps use a private or incognito session to check on each of those without signing out of the main accounts. Since the private or incognito sessions do not save cookies, you will have to sign in and presumably go through verification each time you open a new session; though you may be able to keep several separate sessions going, each signed in to a different account.
Staying signed in seems to prevent verification as long as you do not delete your cookies.
If you have a strong preference for a particular browser such as Firefox, you can install parallel versions such as Firefox Developer and Firefox Nightly which can share bookmarks and extensions, but maintain separate cookies and profiles. There are also alternate browsers such as Waterfox or Pale Moon that might be useful to consider.
You also might be able to bypass the verification entirely by using an old sign-in page like this:
https://signin.ebay.com/ws/eBayISAPI.dll
If after signing in with the old page you get sent to a new sign-in page or verification page, you may be able to skip that by clicking on the eBay logo to go to the main page -- at least that is the way it seems to work for me.
