I have tingling/burning in my mouth when I eat walnuts and I feel sick to my stomach when I eat avocados, could this be an allergy?

Absolutely!! While this might simply be oral allergy syndrome, it could be allergy to nuts. See an allergist for appropriate testing and evaluation to determine what you are allergic to.
Could be. It could be, just a mild form. Everytime you eat nuts if the feeling is getting worse means you need to stop eating them and see your pcp. Sometimes the allergies can progress in time and induce a severe allergic reaction.I know there are some test too to check if you are allergic to different foods and an allergy specialist can help you with that.