Hi Brandstu,
Welcome.
The laser will not turn off during cuts and only turns off when it is finished with the job(M5), so you see the movement lines.
Assuming your set-up allows you to turn the laser ON with M03 and OFF with M05 this problem would generally be overcome within the post-processor you are using.
Feed-rate moves (G01,G02,G03) are cut/engrave so the laser is turned ON. Rapid moves (G00) are positioning so the laser is turned OFF.
So modify your post-processor to add an M03 before each feed-rate move and an M05 before every rapid move.
This system would work and at least it would get you started.
As you will soon discover, the M3/M5 commands are slow to operate so you really need to be using the M11/M10 command set to switch the laser beam ON/OFF. You may also find that switching the PWM ON/OFF is also too slow so you will almost cirtainly need additional circuitry to switch the ModIN on the BB8M between the PWM signal and GND.
Tweakie.