To answer one of your questions directly, no, an electric like the Brinkman does not provide continuous smoke. I'm guessing that you soaked the wood chips/chunks in water prior to putting them in the smoking pan and pretty much they only smoke when there is direct heat from the coils on them. I used to use a combination of chips and chunks in order to try to spread out the smoke time.
Also, you get a fair amount of smoke loss from the unit because it isn't completely sealed...which is ok, you just have to account for that and keep throwing on more wood and it would be good if you could keep track of the temperature swings...it will go cold on you pretty quickly as you're fooling around with it.
I agree with Dennis, though, regarding mesquite. It can get pretty 'acidic' tasting pretty quick on you, and a milder wood might give you more flavor. Just our thoughts.