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...
6 months may not have been long enough for them to work. And allergy shot protocols are not standardized. Your allergist may not have used a large enough dose or the right updosing schedule. I've had shots at three different allergists (I stopped the first two treatments for personal reasons unrelated to the treatment), and this third one seems to give larger doses (which unfortunately makes the shots more painful), and is also the most effective one by far.
It took about a year before there was any noticeable effect for me. It was only at the 3 year mark when I could tell specifically how much better certain things were.
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.