Hi Chris,
a) Please confirm that you are using Azure SQL Database. I believe that is what you are referring to, but there are multiple different ways to run SQL in Azure so I want to be certain.
b) Please navigate to your SQL Database in the Azure portal, Overview blade. What does it show for Pricing tier?
I'm afraid you have selected Serverless, which only makes sense from cost perspective if you only use database for small number of hours per month. The price shown in the portal when you select Serverless shows the cost for vCore seconds, however, it doesn't put any quantity in so in effect it appears as zero cost if you are not familiar how serverless pricing works.
To answer your specific questions:
Setting up a database that actually costs what is quoted.
A: I would recommend you select Basic (5 DTUs) for the Service tier, since your database is really small right now. You can easily increase that to Standard with more DTUs as needed later. For Backup storage redundancy choose Locally-redundant for now.
You need to navigate to your SQL Database in the portal and click on Settings -- Compute + storage blade to change the service tier and other related options as I mentioned above.
What do I do now? I don't really want to pay a bill that has been mis quoted
A: You can open a new billing support request, explain what happened, and ask for refund.
Please create a new billing support request using my instructions below.
Use this link to start the process:
https://portal.azure.com/#view/Microsoft_Azure_CostManagement/Menu/~/support/openedBy/AzurePortal
Fill out Summary, Problem type, Problem subtype, click Next. There will be a delay, then some recommended solutions will be shown. Do not click on any of the recommended options shown.
In the upper left corner, click Return to support request
Click Next to move to Additional details tab, fill out required information, email/phone, etc., click Next, review details and submit case. After submitting you should receive an automated email with a few minutes and a support engineer should contact you within 1 business day.
Please click Accept Answer and upvote if the above was useful.
Thanks.
-TP