What are the symptoms of an oral allergy? Whenever I eat carrots, apples, nuts such as almonds, pears, avocados I get an itchy throat or swelled lips

Exactly that. allergy symptoms can vary from skin rash all the way to anaphylaxis, so what you describe would be consistent with allergic symptoms. suggest seeing an allergist for complete testing and prn meds.
What you described. are the symptoms of oral allergy syndrome considered to be caused by cross-sensitivity between pollen and fruits with share allergens. It is also called fruit-pollen allergy syndrome. Although OAS is not usually life-threatening, it is best to avoid them.