You are very close to the end of the demo. Apart from your occasional grind, I don’t think you are missing much. The first time when you picked up a Special Gun (limited bullets), you have immediately pressed the switch weapon button by mistake. That’s why you thought you have lost the gun for no reason.
Also you have missed out on upgrading the perks and picking other guns in equip menu. It’s almost like a skill tree. I suggest you to try that out. You will get to experiment with variety of weapons (guns, melee, bombs).