Catching small errors in lecture materials
If you catch this during teaching, this means that you probably should rehearse (out loud) and then similarly things should emerge. You already mentioned you rehearse, but maybe you need to rehearse in a more elaborate manner (in a different room, in front of a mirror, etc.) to change your mindset and then find new mistakes. If you rehearse in your head in front of your PC, you are stuck in the same state of mind and won't find mistakes.

Other typical approaches:


Typical proofread advice


Read from back to front and mirror images/figures to get a fresh view upon things. This helps finding misplaced labels.
Spellcheckers and other digital proofing tools (programs which check if borders are met, elements are hidden, etc.)

Teaching notes as bullet points, which you periodically check during the lecture, to ensure you are still on track and have told everything you want to tell (i.e. have not forgotten some points)


These should also uncover any kind of logical mistakes. If you follow your line of thought during rehearsal, this should help explaining correctly and in order.
Notes provide self-confidence, which greatly reduces being error-prone. If you are less nervous, things go easier and mistakes are not so disrupting.

Find someone to proofread (either mutual agreement with someone or pay someone until you are more experienced. A student could be a good choice)


Finally, things can and will go wrong. It is more about how you handle the mistakes. Maybe simply update your slides after the lecture and tell the student next week, that last weeks slides were updated, with comments where things were not explained correctly.