slightly off topic-ish intro
I have a similar issue with fish. I never really liked it (though I’m starting to find some of it ok, lately), and used to even hate it. In the days where veggies were much on the rise and in vogue, I found their reasons to not eat meat a lot more appropriate when it comes to not eating fish - in the latter, a whole eco-system is threatened and species of fish are seriously endangered, where in the former, the ‘only’ objection is the way the animals are treated (still highly objectionable in many cases). Fish-farms mitigate the extinction problem, but the amount of sedatives and anti-biotics that need to be added to prevent the fish from stressing out being too high numbers in a too small basin are probably worse than those already used on cattle.
My general take is that if everyone would eat a maximum of 150 grams of meat a week, no matter what kind, we would all receive all the important ingredients you most easily find in meat, enjoy the benefits of the great taste of meat, and at the same time we could pretty much do with free-ranging cattle meat and no longer need the meat-factories.
slightly off topic-ish intro ends
Anyway, my approach to the fish problem has been to clearly express my dislike of fish once, then repeat as needed when food comes up before it is actually cooked. Generally, hosts want to please their guests and so it is rarely a problem. Besides, I am perfectly happy to just eat vegetables. But in some rare cases, when someone has really gone out of their way to prepare a nice meal and the fish is included, I eat the fish anyway. If the fish is not mixed into the food, I will eat around it, or offer the fish to someone next to me who really loves fish (almost everyone does, so that has never been a big problem).
In short, I would stress that “I’m a veggie, and if at all possible, I would really like to avoid eating meat as it makes me feel really uncomfortable.” (you don’t need to state whether or not that is mental or physical comfort - either is equally valid anyway).
That should be enough. If you still encounter a meat meal, it is a matter of weighing polite gratitude for being allowed to be a guest at someone’s table against the oversight of someone who’s failed to take your expressed preferences into account and your own discomfort with eating meat.
I just imagined that someone could probably bring a nice passage from Ms. Manners on this subject to the table … eh … topic.