So far it's been a few months since people have started using the right pronouns. I feel good using them with people I only met recently or didn't know very well, but especially with close friends whom I've known for longer, I feel uncomfortable. Both when they use them as well as when I do. I'm guessing I'm picking up on their uncomfortableness or something. The language I speak with them in is very heavy on pronouns, meaning I have to use them very often when talking about myself or other people. It can be avoided in some cases, but it's hard to do so and will sound weird most of the time. Obviously, this makes it even more awkward.

Either way, I wanted to ask, does/did anyone have the same thing? If so, how long did it take to properly get used to it?

It kinda feels weird for the same reason. I'm not feeling anything weird for myself, but when they say it, I keep thinking what they might be thinking and how seriously they take it, etc.

Once I knew who was serious and who wasn't, it went away.

It felt weird for me the first few times I heard long-time friends use "he/him" to describe me. But it was mostly a good weird. I agree with TKGW, for me any anxiety I had about it was related to who was taking me seriously and who was just playing along. It was much easier to hear it from people who were meeting me for the first time as a man, because they had no reason to call me anything else.

For the most part, it went away after a few months. I still struggle with my parents. I don't see them very often, and they still struggle with pronouns, so no matter what they use, I tend to cringe.

It feels good and natural when I perceive that person uses them because they are convinced I am truly male, but sometimes I get doubts about who's serious and who's not.

I noticed one of my cis female friends automatically started using male pronouns without me explicitly telling her to start using them, which is great. It's something I appreciated a lot, and I'll tell her when the right moment comes. I remember her telling me she met a trans woman, so I guess she got educated on trans issues.
There are a few others who use them because they saw me correct them, and because I refer to myself with male pronouns around them (I always do this with people I am out to, while I use female pronouns with people I'm not out to, that's my general rule).

Does it feel weird? Sometimes....usually when I can tell they don't do it naturally. I get paranoid they don't take me seriously, and this might irritate me.

I've recently come out too and I realised the only reason I feel weird about it is because I'm hearing both pronouns. I'm hearing some people say she (my parents who are absolutely terrible with pronouns, distant family etc) and my friends attempt to say he. I spent a whole day once with a few friends who practically make no mistakes and I felt extremely comfortable and fine with hearing he, it felt natural. Then I got home to my he she he she parents and that's when I felt uncomfortable hearing 'he'.