A service dog may do the opposite of what you seek, especially in light of the anxiety issue. It doesn't really help a person become more social so much as it thrusts unwanted attention upon the person by the public. Where once you were ignored as uninteresting, or avoided because of the oxygen, with a service dog total strangers would be approaching you, possibly touching you to gain your attention, almost certainly touching and distracting your dog, and asking invasive questions. "What's wrong with you?" "Why do you have a service dog?" "Don't you know dogs aren't allowed in here?" Usually they don't mean any harm, but sometimes they are angry and shout.
Is it all horrible? No. If it was, no one would want a service dog. But it isn't all rosy either, nor is it a panacea. For some people the invasive interactions with the public aren't as bad as not being able to do whatever it is the dog does for them. I personally hate being noticed and approached by strangers. I hate getting lost in the cold without a coat and no clue how to find home or call for help even more. I have seizures and become disoriented. I've spent hours wandering around confused, which isn't as horrible in good weather, but is life threatening when I do it in the snow with no shoes or coat. If a hundred people accost me because I'm novel looking with a dog I'll tolerate it for just one time he safely guides me home. That thing (or more accurately "those things") I cannot do for myself and can be the difference between life and death, makes the public tolerable.
Don't apply for a service dog to make you feel more social or more comfortable. Apply for one if you need help doing something you cannot do for yourself or with other ordinary means.
"Animals whose sole function is to provide emotional support, comfort, therapy, companionship, therapeutic benefits, or to promote emotional well-being are not service animals." -- U.S. Department of Justice, the government agency charged with regulating and enforcing the ADA, and the agency which literally wrote the definition of "service animal" under the ADA