I always walk the execs as well. Only on some holes where there is a long distance between green and next tee do I fall behind. (Like on Churchill between 5&6). I think some people in carts think that walkers will automatically slow them down. But when you consider that I can walk directly to my ball, rather than from the cart path to the ball and back, it all evens out. Plus, when I get to my ball, I have all my clubs with me instead of walking back to the cart when the lie is not right for the club(s) I brought.
IMO, if someone grouses because I walk (and I haven't personally encountered that yet), I would chalk it up to it being their problem, not mine.
BTW, I don't walk the championship courses, mostly because of the distance between holes. One time back in CA, I walked with a twosome in a cart in the first group off in the morning. We finished 18 in 1 3/4 hours. They would be at the green by the time I got to my drive. I think they were playing polo golf. I went home and went back to bed!
On your other question, I don't know of a way to address that without creating tension during the round. I've encountered people who I believe must have belonged to exclusive clubs in the past where there may be few groups on the course at any one time. So expediting play was not a concern. Pair them with some players who are used to pushing and/or being pushed around the course, and you have the potential for conflict. Likely neither one is going to change. What I have done occasionally in those situations is to walk off to the side and slowly move forward out of sight and danger, just so I don't wind up standing still for a long time.
Enjoy your rounds. As my dad used to say, "Don't let the b**st**ds get you down."
