.. Try this. Get 10 mounted Slingers +3 weapons/armor, and pick against you 10 Eastern Archers units, see who wins.. (Try to use the Slingers in a way that what I mentioned above is applied).

Also, when using Slingers against blocked stacks of Infantry, it's better to assign your Slinger to attack the last legion/infantry unit, to make the most use -again- of what I mentioned above.