Skipping straps is fine, but your progress might take longer. I prefer chalk when I can use it. My gym complains when I do.
how is your set up? where are your hands on the bar? maybe that is part of the grip issue.
Mark Rippetoe on set up.
look at some vids of good pulls to see other peoples sets up. and where there hands are in relation to their feet.
I use the overhand grip, and then the mixed grip as I get heavier. You can switch up the hands during sets or week to week as
suggested ealier. Someone mentioned else the hook grip, I use this too it helps me too and its helped my cleans and snatch.
For a long time I used a mixed monkey grip like in jujitsu or judo-thumbless maybe someone who knows better will tell my why thats not a good idea and newdamage really nailed it
dont cheat yourself. the grip is the weakest link. My grip is good, with chalk I have much better pulls.
Grip training- I used to do towel chins, and plenty of pullups.