At the moment I can't pick a welcome email or double opt-in email from my content builder content, i have to pick from classic manual coded HTML emails.
Is it possible to use the content builder content for those operational emails?
Currently, using the 'welcome email' option on a list only will allow you to access the classic editor emails. It is in the works to get this updated (similar to user initiated and trigger sends) to allow for content builder emails - but I was not given an ETA on this update.
Your best bet to use a content builder email in the meantime is to set up either an automation with a filter to periodically send out a welcome message (scheduled automations can run hourly at most) or to have your sign up form make an API call to trigger the message to be sent from SFMC.