LED ULN2803 thermal / heatsink questions
If you are driving your Darlington transistors into saturation, then the \$V_{CE}\$ is a maximum of 1.1V according to the datasheet. So, the power dissipated in the transistors is 1.1V times the current rather than 5V times the current. The temperature increase is proportional to current, so you have about a 30C rise, which is pretty manageable.

If you are time-multiplexing the LEDs (what you call a round-robin) faster than the thermal time constant of the Darlington packages (maybe a second?) then the average temperature increase will be proportional to the duty cycle times the power consumption.

I think the Darlingtons will get noticeably warm but a heat sink is probably not needed.