I made the mistake of using some extremely diluted Dawn in my Neem spray and it still ended up causing some rather bad burns on my shishito plants.
Do the oil-based soaps like Dr. Bronner's not have such issues if used in moderation?
Dr Bronner's is the only one that I use. The sunlight is the main problem. Your leaves have to be completely dry before the sun comes up.
Emulsify at a ratio of 2 parts neem to 1 part soap. I get almost zero suds in my sprayer when I use this amount, so that tells me that I've got it right.