Congratulations on the new tow vehicle. Something about a new truck just always seems to feel sweet. The push pull thing you mention is normal, and you get and feel it even when you aren't towing, and I've always found it to be more noticeable in a larger empty pickup than a smaller pickup or car. Should you be in the slow lane empty for some reason you can tell if a truck is coming up on you without having to look in the mirrors.
While I'm not in the camp of thinking pushing the tow vehicle hard shouldn't, or can't be done, I confess to preferring the having a vehicle somewhat larger in rating than the task requires. Having a little extra power is nice, and of course the more solid feel, and things like the "Mirrors" that are matched to the task at hand make the whole experience all the better.
Sounds like you're going to be a "happy camper" for years to come.