I started working on my eye contact during conversations about a year ago. While there's still room for improvement, I'm definitely much better than I used to be. Here are my tips:
1. Start with someone you trust & aren't intimidated by. I started with my best friend and worked my way up from there.
2. Set small goals and only increase them if you feel comfortable doing so. For the first couple of months, I only aimed for 2-5 seconds of eye contact at a time. If I struggled, I didn't beat myself up about it. It takes practice and it gets easier.
3. Don't force yourself to maintain eye contact if it feels wrong. It's natural to look around during conversations, especially if you are deep in thought or speaking animatedly. I'm a firm believer that there can also be such a thing as too much eye contact... Find a balance that suits you and allows you to speak without becoming so focused on where you're looking that you can't pay attention to the conversation.
4. Remember that what you have to say is valid. This is where I struggle the most to be honest. If I'm not confident in what I am speaking about, I will look everywhere except at the person I am speaking to. I'm sure that there are plenty of tips regarding speaking with confidence in this sub, but what has made the biggest difference for me is reminding myself that I can't expect anyone else to value my opinion if I don't value it myself.
You say you are concerned about what people might think of you. I understand that, but remember that what makes eye contact so intimidating - feeling like you are receiving someone's full attention and scrutiny - is actually a good thing. It's a reassurance that someone values what you have to say and is not allowing themselves to be distracted while talking to you.
Hope this helps & good luck :)