yes, i always take the resivior off if its dirty, like really bad, and soak it in some brake cleaner and what not till its clean, then put it back on, fill with new fluid.
If you can find some blue ATE fluid it helps BIG TIME.
I always push a little bit of it into each of the lines then suck it out of the resivior, then go at it, soon as i see the blue go away i know its 100% new fluid in the lines.
Protip, you cant buy it anywhere as its banned by the FDA because it was blue.
I have 12 cans of it left and plan on keeping it for life. I use about 1 can every 5 years as i use it sparingly and not as a full system full fluid.
Seeing as you ran it dry, you want to bleed that suckweed with pressure for sure. I never use the pressure systems the way they were made to be used. The problem is that brake fluid absorbs water even from the air. So i never ass the brake fluid to the tank. i just use it to pressurize the system. make sure you refill the resivior every time you do a side. you dont want to let it go low.