Never tried dogs. I did grass allergy shots for about 6 months as an adult. All that ever gave me was an itchy arm. My nose -still- becomes a spigot every time I have to cut grass...
It it definitely not a quick fix, you need to commit to the long haul.
Edit Yeah, doing out of pocket definitely makes it harder to keep up with. Fortunately my insurance covers most of the cost of my shots.