A little more info, please. Are antihistamines not working for you? Which ones have you tried? With the cat and dog allergies, I strongly suggest Zyrtec. And have you tried anything for the reflux? It can certainly cause this type of pressure and shortness of breath.
With the few allergies you have, I'm not sure the shots are worth the effort at this point. And it's not months, it's years - 5 is the normal regimen. So that's quite a committment for this. If you try the various meds and they don't work, then maybe shots are the best route.
As for myself, I was allergic to all the standard allergens tested for here...and all at a 3 or higher (with most being 4 plus!). I've been this way all my life (61 years). I've done the shots 3 times (teens, 30's and 50's) as they don't last forever and allergies do change (Dogs used to be my worst and now they are way down the list...cats were #2 the last time I had the testing).
So, bottom line, I'd work my way thru the meds and see if you can get a combination to work for you rather than try the shots right now.
Good luck!