In response to your post, I will do the best I can, but more experienced beekeepers hopefully will reply too.
Firstly yes those full deep supers are terrible unless you are a very strong person. I started my beekeeping with them also and quickly understood they would kill me. I slowly transitioned to mediums, which is all I use now. Much better having all the supers the same size also, so you can move frames around where ever you need them.
Crushing the queen is always a fear I have also when doing inspections. But I remove the frames very carefully and look on the frames both sides before I move them anywhere. If you move slow and carefully, you will not crush too many bees.
On that top super that is so full of honey, what is below in the bottom super?
Hopefully the queen is down there with the brood and hopefully there are some frames where the queen can lay. I would make sure the queen has room to lay firstly so they do not swarm on you. If you check down below and there are empty frames in there (most of the time the frames on the outside are empty - meaning frames number 1 & 10), place them between frames filled up with brood to open up the broodnest and give the queen more room to lay.
Now you have to address that top super filled up with honey. I would place an empty super below that super. It could be a medium super if you have any. If you do not have any empty supers, or any empty frames, well you have the option of extracting a few frames of that honey and placing those empty frames back into the honey bound super in between frames of filled honey. This at least would open up the honey super until you receive more of your equipment.
What is happening with the other hive?
I did not understand from this post. Perhaps you can transfer some frames into the other hive if they are not honeybound as well.
It sounds like you have 2 good queens and so I would not worry about the hives. You never, ever have to see the queen. All you are looking for is either eggs or larvae. Sometimes I just stop my inspection when I see a good brood pattern.
I hope I helped some. I am also still pretty new at this and just sort of move things around by vibe when I visit the bees. I know I always have to watch that the brood nest is open enough for the queen to have room to lay. I know you do not want to have a ceiling of honey above the broodnest or they may swarm.
If you had honey dripping down onto your boots, then you were a bit rough in handling the frames and broke the seal on the capped honey. That is why the bees were excited and buzzing around you so much. Make sure you do not hold the frames sideways if they have uncapped honey in them. The way the combs are shaped, well the honey drips out easily that way.
Take care and keep asking. The answers will come