There are limits to the number of times you can sign in and out per day using the eBay app. It sounds like you may be running into those limits.
If you have more than one account and you are signing out of one to sign in to another repeatedly, you might be able to avoid signing in and out so much by using the app for one eBay account, and using a different browser for each other account. Or a different private or incognito browser window for each account. That way you can stay signed in to multiple accounts at the same time and quickly check messages or sales without needing to juggle sign-ins all the time.
