BrianCL
If you refresh the page you're on (usually with the F5 key) immediately before you select the link to connect to your developer environment you should be able to connect successfully then.

While you can have several developer orgs, to collect your badges and achievements all in one place you should initiate each challenge from the same Force.com / Trailhead account.

So first log on from your preferred account and launch all challenges from there.

When you launch a challenge it doesn't matter what dev org you link to in order to complete a challenge.

Points are applied to the Force.com / Trailhead account you're logged into at the time when doing a challenge. If you're unsure how this works complete a module that has a Q&A type challenge ( i.e. Doesn't need you to link to a separate developer org and you'll see the challenge points awarded to your account. )

Someone else here may be able to provide a less adhoc solution to the page refresh issue.

One other thing you might notice is sometimes when you initially logon with your preferred account for storing the badges and achievements they may not be initially up to date - It does get updated after a few minutes, sometimes only after you go into a module.