Good day,
... 10 USD per DAY...
Yes... according to your description this sound high
The website is not live yet, and I'm the only person using it in test
Since this is not a production database yet, then:
(1) monitoring the issue should be simple.
1.1 Activate Extended Events and monitor what and who is happen in your database. Monitor queries executions and if you use Auto-pause then monitor when the database wakeup
1.2 Check this doc about monitoring the database
https://video2.skills-academy.com/en-us/azure/azure-sql/database/monitor-tune-overview
Check this as well:
https://video2.skills-academy.com/en-us/azure/azure-sql/database/intelligent-insights-overview
Note: remember that the cost for a serverless database can raise above the cost of simple Provisioned database per seconds when both are active. serverless is good if you have big differences in uses (unpredictable usage patterns) like case where most of the time the database is not in-used. Test the option of using other tiers like Basic or S0 maybe.
(2) Check if your cost it coming from specific database and not from the replica or other services like backup, security and so on. Remember for example that Zone redundant database will cost about twice.
(3) Check if you have Long-term retention
(4) re-check that your pricing tier was not changed to Business Critical service tier which cost about 3 times more (I had a client which had this mistake)
(5) try to enable Auto-pause (it makes no sense that you work 24/7 and since only you use the database, this might fit your needs). If you did, then remember that if someone execute a simple query like "select 1" every minute, then the server might stay in active mode and your cost get very high even so there is almost no actives.