I am the exact same way, and I understand entirely how you feel. I am both introverted and shy (and discovered recently that those are NOT the same thing.)
It takes me a good 6-10 years in a ward to have friends. I go to ward events and stand there dumbly, too paralyzed to initiate conversation, and hoping someone will help me be included. I usually get friends after a lot of years, only because I force myself to participate in callings and service. So I naturally meet people when I visit teach, or plan a lesson, etc. It just takes me a long time.
I second the person who suggested that you do activities where you can work alongside others. Sitting at a party and engaging in "small talk" just doesn't work for some folks. But working alongside someone gives you a common goal to discuss. You are not "on the spot" to discuss yourself and can focus on the project. And people naturally come to love those they serve with.
I recommend you read a book called "Quiet." I just read it, and it is EXCELLENT!" It totally changed my perception of myself. One of the things it mentions is that shyness and introversion are two different things. Introversion means we generate our energy internally, and feel our energy drain away when we interact with others. Extroverts are the opposite. (They feel energized by other people, and bored and not their best when alone too much).
Shyness is a form of social anxiety. It talks about how introverts may reduce our social interactions to preserve our energy, as above. But the reason so many of us are also SHY is because being an introvert is disvalued! Extroversion is such a valued trait in the United States (and some other places) that we internalize the idea that something must be wrong with us. Then THAT makes us feel insecure and feeds our shyness further. It need not be that way. For example, many Asian cultures value introverts more, and find extroverts "talk too much."
) The point is, introversion and extroversion are just two ways of being. They are neutral in terms of being "good" or "bad." So stop beating yourself up about being an introvert. There is nothing wrong with that!
Realize that not all introverts must be shy. You can work on your shyness and find ways to make friends. I know it's hard. It TERRIFIES me! Ironically, the book quiet mentions that introverts are often the MOST active and self-revealing online. So it's not that introverts have nothing to say, or can't share. It's the face-to-face that's hard.
This is what works for me. 1.Give it time, and don't beat yourself up. It's OK not to have tons of friends. One or two is enough, and you'll get there. 2. Pray for help to know whom to befriend, and how to do so. 3. Sign up for any project or service you can. Accept callings. 3. Go to the parties, even if you don't talk. If you leave after 5 min it's ok. If you have a spouse, bring your spouse. My husband helps me talk and engage. He knows me and can bring things up that include me. If I just can't bring myself to go, that's ok though. 4. I try to stop focusing on finding friends for myself, and try to find someone else who looks lonely. Serve him or her. Somehow I can be more brave in extending myself if I think "I don't have to reveal myself," but so and so needs help. Friendship and sharing of self can come later. Just look for others with a need and fill it. They will come to love you and will want to know you too. 5. Did I mention pray? Pray. Pray. Pray. 6. Journal. Everyone needs a safe place to be themselves. When you don't have friends, it can help to pour out your feelings in a journal. Then nurture yourself-whatever is nurturing for you. 6. Do visiting teaching and home teaching. 7. Get to know your neighbors on either side of your house. Whatever is happening at church, it's important to have that sense of community. You don't need to be best friends. Just bring them a treat, explain that you live next door and want to know them, and go from there. You can do it! 8. Don't believe people who imply you are at fault, and somehow defective. We all make friends at our own pace. 9. Don't allow your social fear to keep you from church. It's easy to get offended or discouraged or want to give up. Don't do it. Even if you don't speak to a single person and are not acknowledged by anyone at church. Christ can be your best friend. We NEED that relationship. Build on your personal relationship with your Savior. Listen to the Holy Ghost. He will help as you seek friends.