You may also want to cost/benefit it.
As others have said, the convention itself is essentially one massive crowd; attendance for the past several years has been in excess of 60,000 attendees. There are some quiet corners to be found, so to speak, but the ICC hallways, attached hotels, nearby restaurants, food trucks, etc. are going to be packed.
You'd mentioned your interest in Dungeons & Dragons and, in my experience, a lot of the tabletop RPG games are held nearby hotel conference rooms where there will be maybe ten round tables of 5 - 8 or so players, each running at the same time. If you're into Pathfinder, its official events take place in a massive ballroom with literally a hundred or more tables running. Crowds and new people do not at all bother me, but even I give the occasional sigh of relief when getting back to my hotel room and having a few minutes of solitude.
4-day badges will run you and your wife in excess of $200 and if you are staying in Indianapolis, hotel and meals will easily run you up to a thousand dollars or more. Many attendees spend several times that depending on purchases in the vendor hall and event prices.
I've been attending the convention for roughly a decade and it is absolutely one of the highlights of my year. I'd encourage almost anyone to give it a try. However given that you have a "lot of social problems, issues with meeting new people and being in large groups," please bear in mind that the nature of the convention itself means that you will constantly be exposed to large groups and in almost any roleplaying event will be meeting a half-dozen or so new people.
I don't want to discourage you from attending, but it is a significant time and financial investment and it seems to me that environment at Gen Con is one that will significantly test your social problems and issues on a regular basis. If I were you, I might do the old trick of writing down the pros and cons and seeing how things stack up.
Also while Gen Con is absolutely incredible, there are also many other smaller, regional gaming conventions that might be more conducive to your particular situation. You could easily attend two or three of these throughout the year for the price of a single Gen Con and possible have a much better time with much smaller crowds and so forth.
Regardless of your decision, best of luck and I hope things work out well for you.