I have a bunch of SQL databases. How should I put these into Azure?
the first obvious solution could be the adoption of a SQL Server VM hosted on azure.
You can move there databases and ssis packages in any traditional ways.
This is an Azure Iaas solution and usually is the most expensive. You can reduce cost moving your on prem sql license wiht the &quot;Bring Your own license&quot; solution coming from software assurance license mobility option.
But you can also adopt Azure Paas solution using:

Azure SQL databases (https://docs.microsoft.com/en-us/azure/azure-sql/database/)
Azure SQL managed instances (https://docs.microsoft.com/en-us/azure/azure-sql/managed-instance/sql-managed-instance-paas-overview)

I suggest you the second one because it is almost equal to the a on prem sql server instance. But both laks of ssis solution.
You pointed right to Azure Data Factory to relace dtsx packages but you can also run the same dtsx packages inside azure data factory:
https://docs.microsoft.com/it-it/azure/data-factory/create-azure-ssis-integration-runtime
In the meantime you redisign the ssis packages in ADF you can eventually run them temporarly in a Azure VM.
For moving database to an Azure SQL Server Managed instance i suggest you to use DMA:
https://docs.microsoft.com/en-us/sql/dma/dma-overview?view=sql-server-ver15
You can do a online or offline migration. Look at this:
https://azure.microsoft.com/en-us/resources/videos/data-migration-demo-using-azure-migrate-dma-and-dms/
Here you can find a free lab to test the procedure:
https://github.com/microsoft/MCW-Migrating-SQL-databases-to-Azure