Meat allergies seem to be more of a problem than many people think, especially doctors.
There is a meat allergy article on this website that gets quite a few visitors.
If you have a meat allergy, then it can be managed by avoiding the specific types of meat you are allergic to. Many people report being allergic to just one or two types of meat. Occasionally people are initially allergic to one type of meat (for example pork) and later develop an intolerance to beef.
There must be other people who are sick a lot (or occasionally, depending on their eating habits), who do not realize that they are allergic to meat.
In this case, it is well worth cutting out meat for a while to see if you stop getting sick.
Some people seem to need at least some meat in their diet, so if cutting out meat prevents you from getting sick, you can then try specific types of meat to see it’s effect:
See the list of meat for ideas on alternate types of meat. This may give you some new ideas about what to eat.