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I used to be there, and sort of figured palm muting just wasn’t my cup of tea. But then I decided I would try anything Tony threw at us. So the next time it came up in a lesson, I worked out the best position for my hand up against the bridge. It might mean shifting your hand slightly during the stroke in order to affect all the strings. It is a bit awkward at first to keep the palm on the strings while holding the pick, let alone move it up and down, but it is a skill you can learn.
As far as the pick getting “stuck” on up-strums, what I found is to slant the pick in such a way that it doesn’t “catch” on the strings. That means changing the slant on each down- and up-strum. You have to do this consciously for a while until it becomes a habit.
Hope this helps!