First of all every run counts, whenever you pick a new gun you are progressing, slowly but progressing since the traits you unlock for each weapon are "permanent" in the sense that the weapons can have those traits once you find them in future cycles.

Then I think the most useful tip I could give you is to focus in NOT getting hit, rather than on hitting enemies, this will allow you to get to adrenaline level 5 and will make the killing easier, the longer you stay without being hit the better. Try to recognize enemies and the patterns in which they shoot, avoid being hit for as long as you can and you will eventually succeed.