I think if you can't have gas, you should probably have some other local anesethic, because even though the cavities are shallow and may not involve as much drilling, you might still feel some pain, esp. if you are really sensitive to it. I have that particular problem even with shallow drilling in the enamal, I still feel pain in the tooth.
Now a days, the shots aren't so bad, really, as there seems to have been some changes in the past few years, I've noticed. They usually dab the area of the injection with a local, which tastes funny but takes a lot of the sting away. Depending on whether it's the upper or lower jaw, they'll either inject near the teeth(upper) or near the back of the mouth(lower jaw where all the teeth are connected with the same nerve). The lower jaw is a bit trickier and might need more time to numb because they have to block the nerve usually a distance away from the tooth. Usually, on the upper jaw, it might be 1-2 injections.
Then my dentists tests to make sure the anesthetic is working properly and doing its job. If not, then she gives another injection to be sure. For the lower teeth, I have to have the longer lasting shots which leave you numb for up to six hours, but it's different for everyone.
It's very important that you're catching these cavities early. Less drilling and filling so it shouldn't take that long at all. My dentist did one small cavity and replaced two old fillings in about a half an hour, after I was numb. If you don't take care of the fillings, then down the line, you might be looking at the more extensive dental work like root canals which are much more prolonged in terms of time spent in the chair, so it's good that you're being pro-active.