Good, you have your meds.
Avoidance of peanuts and tree nuts is going to be based largely if you have any tolerance for cross-contamination or not, loosely called "may contains". It is perpetually frustrating because there is no law which requires manufacturers, suppliers, etc., to disclose manufacturing information including shared production lines and associated policies which include deciding for consumers whether their cleaning process eliminates all risk (hint: it does not).
To not overwhelm you yet provide a concise indoctrination, all items will be on a continuum of low risk to high risk based on your risk profile a.k.a. "comfort zone" which is affected by manufacturer products and practice.
Read every label every time.
Call the manufacturer if you are at all hesitant. Be prepared to hear a scripted and unhelpful response.
Ask here if others have tried it with success as a one data point of many.
You'll eventually find a semi-totating list of manufacturing practices you're willing to take risk to consume.
Gluten is tough, but probably tougher in a sense for me than you. If it's inly the gluten proteins and not the others you need worry about and it's an intolerance digestion issue the you could likely go by package claims to be gluten free. I have a child anaphylactic to wheat, barley and rye in the same manner my other is allergic to peanut so we must avoid all proteins not only gluten and it is not a digestive issue but involves the same IgE mechanism.
I'm still not super sure where you're at on "gluten free" as an intolerance because allergists don't diagnose intolerances and will forward that on to a reputable GI specialist for investigation of Celiac. Nonetheless, to play it safer for you I'll assume an actual allergy. Others here deal with Celiac and are going to be able to sort that out better.
ETA: Definitely look in the sesame subforum due to its status as an unregulated allergen in the US.