Q.Should I use insecticidal soap or Neem Oil on my vegetable-flower gardens?
I have leaves all over my vegetable and flower gardens with holes in them. The holes are mostly around the center seam and then the leaf turns yellow and dies. I have found green worms (cabbage worms, stink bugs, red spotted spider) and others. However, none of them are in great numbers, yet. The research I’ve done suggests that it may be spider mites. I even took a white sheet of paper out to the plant and shook the leaves over the paper and look at it through a magnifying glass and saw little bugs running around. I have also noticed a few webs on different plants. My Rose bush has the most leaves like this. So. . .
1. Did I diagnosis the problem correctly and 2. What should I do? I really would like to go organic, and I also know that the soil was nitrogen depleted (not from a home test) and the plants are most likely susceptible to a lot.

Certified GKH Gardening Expert
Various types of worms can affect garden/vegetable plants. Those that cannot be picked off can be treated with products containing Bacillus thuringiensis (Bt), a natural bacteria that kills worms without harming
plants. Neem oil may be helpful as well. Neem oil is also effective on many pests and safe to use on your plants. As a bonus, it is also a great fungicide. Here is more information: https://www.gardeningknowhow.com/plant-problems/pests/insects/prevent-caterpillars.htm
https://www.gardeningknowhow.com/plant-problems/pests/pesticides/neem-oil-uses.htm