I would say between the two, karate. Your head will get shaken with judo as it's a very physical style with a lot of grappling.
If you choose karate, make sure you talk to your instructors first and make sure they know you don't necessarily want to spar.
It sounds to me like you would benefit a lot from the basics, the exercise, fundamentals and kata.
When I'm in a bad way I practice kata at home. It really breaks up stress.
It's perfectly normal that you want something slower paced.
There's three types of martial artists:
The Student- always learning, forever in love with acquiring more knowledge and perfecting the form. The most common type.
The Teacher- able to retain everything they are taught, and is talented at teaching others. A rare type
The Fighter- a freak of nature. Their talent isn't necessarily retention of knowledge or perfect form, but moreso that they are experts in application. Extremely rare.
Let me know if you decide to go and take a look! It would be an interesting thing to see posted, a beginner's journey through martial arts.