Azure DB paused and won't unpause

The Azure SQL database is designed to auto-pause when
a) it hasn't been active for a period of time,
b) when it reaches the limit of a resource.

For the first, you just attempt to connect again and it wakes up on its own.
For the second, there is an option you set to continue, meaning that you choose to incur charges if it goes over the resource limit.

Mine has the 'continue' option set, and yet it has paused because it has reached its resource limit.

Now I can't wake it up. The application just times out.

There doesn't appear to be an option on the console to un-pause since it is supposed to happen automatically.

There are a number of triggers that will resume an Azure SQL Serverless instance such a performing a login, copying or exporting the database, etc. You can find a full list in the official documentation here.

In your case, where it seems that attempting to login generates a time-out, you can force the restart of the server by manually scaling the database up and then back down again.