My organization has Office 365 Enterprise E3 licence and it's my understanding that you can use Powerapps to run and create apps that do not use premium connecters. However, people are getting messages to extend their trial.
Am I able to keep using PowerApps to create/run apps with non premium connectors>
I think that you are able to use PowerApps with nonpremium connectors.
From the documentation about Pricing, we know that if we are using PowerApps for Office 365 and Dynamics 365, we are able to connect to cloud-based services like Azure SQL, Dropbox, etc. While we are not able to use premium connectors like Salesforce, DB2, etc.