Unexpected database charges

It would be useful to have more information regarding what exactly the charge is related to? If you don't know, there are a few places where you can check it:

• either in the Billing Dashboard (billing console https://docs.aws.amazon.com/awsaccountbilling/latest/aboutv2/getting-viewing-bill.html) - you can click on RDS and see which AWS region the charges are coming from, and then you can investigate further to see which instance types / resources this is related to
• or, you could access Cost Explorer from the billing console, and effectively use convenient grouping & filtering options to find the charges you are concerned about

Which charge exactly was unexpected? Did you expect to stay within Free Tier? If that's the case, you definitely can run one EC2 instance, and one RDS instance, and you get some free tier limits for data transfer charges. However, it's important to make sure that you are using the resource types that are specifically eligible for Free Tier: https://aws.amazon.com/free/

I suggest to double-check whether you use the resources that fall under Free Tier or not.