These thoughts are held by many people I'm sure, including myself.

I like to have easy going relationships too, yet there is a side of me that wants to find someone more profound and sensitive like myself to talk to about more serious topics without them being afraid. Not extremely deep things, but stuff like being depressed now and again or having anxiety.

I think you'd be surprised how many casual friendships can easily become much depeer if you two click soon. For example, I made a casual friend who I got along with right away a few months ago. We haven't spent too much time together from various reasons, and yet we both feel comfortable admitting to one another about having anxiety. So we did have an instant connection, and didn't even need a lot of time to share that between us. So sometimes, you might find good relationships like that where it just feels 'safe' to share. Other times, you have to test the waters, feel it out, and see what kind of person they are before revealing too many heavier things. It's all relative.

Self-help group is a good idea. You could try that once in a while. But, at the same time, I'd maybe avoid getting too-too close to anyone there, especially if you find that people you chat with have even heavier issues than you, obviously. I'm not being cold, just suggesting this because I've had a similar situation crop up for me. Did a self-help group thing and connected with a girl with very similar but later I found out even worse problems like mine. She eventually became too attached to me, and we had an unstable relationship. So going to a group like this can't get too deep. Unless you find that both of you have different issues, and can function in a normal relationship without becoming too involved in just your issues bringing you together - which was somewhat like what happened to me and my group friend.