yes - you need 3 teamsters offices.
also, you don't need 2 docks, as it's a complete lottery which ship lands where, and whether the dock workers are there etc.
important is a road link from the mines to the dock.
As Brf points out, more docks do help.
It is not a "lottery," as in gambling, where the ships go. They go where there are goods to be picked up.
The dock workers attend each dock not randomly as in a lottery, but by the rules of employment and personal satisfaction of all Tropicans
The placement of the docks as related to the distance the teamsters need to walk from the mine(s) plus the volume of goods at each dock is critical.
Yes, roads may "speed" the teamsters.