In our current setup, we have a top level TFS project and all the projects within in as separate folders with their Dev and Main branch.
To start trying VS Team Services, I am looking into migrating just a folder(Project) from TFS 2013 on-premise project to Team Services. Can I do that?
The options I am getting are to choose the project collection but not the folders within it? Is there a workaround for that.
It is not. You need to migrate everything as is.
If you want that level of control you need to use the TFS Integration Tools. They are a lot more configurable, and complicated to boot.
I would recommend that you move with only the head/tip.