Hi everyone!

I'm entirely new to the piano, starting at age 18. I have had experience with music, but I'd like a little bit of advice to make sure I'm starting out on the right foot with piano.
First, let me give a little background. I learned clarinet privately and played in band for 7 years on a fairly advanced level as well as participating in choir and one-on-one theory lessons, but dropped out of it during high school. I've been playing handbells and singing in choir at my church for several years, but over all I'd say my musical ability has atrophied since giving up the clarinet, so I'm *mostly* looking to build it back up again and become more able to read music written for handbells, choir and piano.
So, I pulled out the old keyboard my mom used to use and bought a book called The Piano Handbook by Carl Humphries. I think I've had good results so far - inside of two weeks I'm able to play independently with both hands on pieces like Greensleeves and "Morning Hymn" in the book, something I never thought I would be able to do, and I'm entertaining thoughts about someday buying a real piano.
But, I'm a little concerned about the way I'm learning. I'm fluent in the treble clef from clarinet, but even years after I stopped playing it I still find myself finding it difficult to associate the notes with something different. When I look at middle C on sheet music, I don't think "C", I think "left-hand three fingers and thumb down", as it would be on clarinet. This slows down the recognition for me. By the time I learn the piece, I find that I have memorized it and don't even need to look at the music anymore - but to me, this does not seem correct, since it precludes any possibility of sight reading new music that I might come across at, say, one of the church's pianos.
Please let me know if this is a normal/acceptable way of learning a new piece for a beginning pianist. I'll go slowly through the piece until I'm fluent in one hand, and then I'll do the other one (the bass takes much longer since I'm nowhere near as knowledgeable in bass). But, I can't really play the piece well until I entirely commit it to memory. I cannot sight read for the piano. After "learning" the music, I only look at the notes in order to remind me how long each note is. I know I'm still extremely new at this, but I'm worried that I'm never going to get away from this tenancy. Is this normal or should I be doing it a different way? I hope I'm expressing what I'm trying to ask properly, because it's a little hard to explain. I just want to make sure I'm learning the correct way so I don't make it harder to change farther down the line.
All of the piano players I know have had it as their first instrument, so maybe I could get some perspective from others who started out on something different and know where I'm coming from. Like I said, I'm happy with the progress I've been making as far as actually being to play, but I want to make sure I'm on the right path as far as correctly reading the music.
Thank you!
