So far there have been no answers from teachers, and I don't know if any "old" people have answered (I know one is young, but I don't know the age of the other.

So here is an answer from someone in both camps. I'm not young (celebrated my 68th birthday yesterday), am still a learner, also a trained teacher, and have had a foot in the door in piano / music teaching via music teachers.
You are absolutely on the right track, wanting to start with a teacher to get the physical part of the piano. Re: "hands" - also remember that your hands are attached to your arms, which attach to your body. You're at an instrument that is bigger than you, where you're the one moving around on it.

When getting a teacher, it might be good to state at the onset that you are there to get the skills and knowledge you need to be able to play the piano. Experience shows that we cannot assume this; it is often assumed that adults want quick results for playing their favourite music, want shortcuts and such - you don't want to end up being given "olden goldies" to play because that's what you must want.
The fact that you play / are learning clarinet gives a couple of things. Obviously you're familiar with music at some level, understand notation to some degree in the treble clef, things like time signatures, key signatures, counting, maybe phrasing. All this will help you. I play several instruments: recorder (descant, alto, tenor), studied violin a few years, and sing, and ofc piano or I'd not be here. Wind instruments are different than piano. So you have to enter the mindset and body of a pianist. Some differences:
- on clarinet, your hands stay in one place. On piano, any finger can play any note, and your hands move all over the place. (Be careful about books that have "C position" or any "position", esp. if they train your hands that "the thumb is on middle C, esp. for a long time).
- For a wind instrument, a note sounds for as long as you blow. On violin, the note sounds as long as you keep moving the bow. The piano is essentially a percussion instrument where the blow of the hammer which is thrown at the string causes the string to start ringing and immediately decay. You'll be tempted to "hold down the note, keep pressing" for a sustained sound. Besides the fact that the pedal can take over, this will create tension in the hand. In other words, the piano is not a clarinet, but you'll be tempted to relate to it as though it were.
- The bigger point is that your teacher may think you know more than you do, without being fully aware of these differences. I've run into several fellow students who got caught out by this, when they played another unrelated instrument.
You will have a good feel for the melodic side of music, since you play clarinet. You may hear the "singing" part of the music. How to physically make it sing, that's the technique part. Chords and harmony will not be much developed, and this is a large part of piano playing.
I'd probably stay away from any book written for "adults" because often these are shortcut books, designed with the idea that adults want to go fast, and less in depth.