It's hard to be happy and fun when you are sleep deprived. And that infant stage isn't very fun when it limits your freedom and all you are doing is taking care of others and not yourself.
Date nights outside the home may not be possible until your youngest one gets to be a little bit older but that doesn't mean that you can't have a romantic date night at home after your children go to bed. Light some candles, order take out, rent a video and let your husband rub your feet and take care of you for a change.
Exercise definitely will help elevate your mood and, the more in shape you get, the more empowered and proud of yourself you will feel. Find a gym that offers childcare and use it at least 2 or 3 times a week, if only for a half-an-hour at a time.
Do you have a mom's group near where you live? If so, join it if you haven't already and start joining in on some of their activities. This may be enough of a social outlet for you for the time being so that you don't feel quite as stir crazy.
And, lastly, don't be so hard on yourself. I think each one of us here have felt similar feelings at one time or another and most probably, more often than we would like. I know that for me, I am a definitely someone who needs my fair share of personal freedom away from the kids and even away from my husband. I get grouchy and cross when I am not taking care of my social and emotional needs so I have learned that the best thing I can do for my family is put me somewhere at the top of the list. It's not easy but achieving this balance of personal life vs. family life is really the key to my whole family's happiness.
I hope this helps.