There's a glitch where I can't move when I pick up a new gun until I shoot it. definitely detrimental when I run out of ammo in a group of enemies.
Sounds like you are defocusing the game window when you press the pick up gun button. If you are playing on firefox shift - right click will do just this, so sprint with space on that browser, or use a chromium based alternative.
If you are not using firefox, are you playing on one of the recommended browsers from the description? Also make sure you don't have any extensions installed which bind an action to right click/alt or some control key + right click/alt.