Does SQL Server redistribute data after adding an additional data file?
The way that datafiles work, any new data will, mostly, get put in the new data file you added. Or in your case the two new data files.

However, SQL Server will not try to re balance your data. This would't make sense either as you will be causing a lot of IO / load on a database for, as far as SQL server knows, no reason.
You can however rebalance your data by recreating your indexes on a new filegroup while dropping the old indexes. The advantage being that you're allocating space in small chunks, getting unfragmented indexes and generally have more control.

You will however have to do this largely manually. And you'll probably want to create full backups before and after each operation.