You say on the second pass (second shave) the razor is gripping on your skin. I would bet that what's happening here is that you're loosing 'glide'. The purpose of lather is to soften the whiskers, but also very much to provide cushion (via the tiny bubbles) and glide (lubrication for the blade to glide across the skin). The correct amount of water in the lather allows for glide. A 'gluggy' lather (too little water) causes the razor to get 'stuck' on the skin (it can dig in, too).

I'm pretty sure what's happening is your lather is drying out between passes. You probably take a reasonably long time to shave at this stage as a you're new to it all. there's nothing wrong with taking your time of course. For me, it takes me ten minutes total including making the lather, stropping, shaving, stropping post lather and cleanup; you would likely take longer.

Even though I can do a single pass (as indicated above) in ten minutes, if I'm doing two passes (which is quite rare for me) I add water to the lather mix to weaken it a little. I think this is really important especially for that ATG pass. Also, often, after applying lather to my face for the FIRST pass and then stropping (30/70) while the lather sits on my face I'll often dip the brush in water and go over my lather again; as it may have dried just a little on my face.

In essence, you need to ensure your lather has enough water in it. Not TOO much or else you loose cushion, but enough to provide glide.