Honestly, there are no secrets to it. It takes time getting used to suckling on a pacifier, there is no trick to using a pacifier. It takes time and effort. It might feel uncomfortable at first but after awhile, you will be at ease with it.
Maybe you need a less stiff pacifier. The one from awwsocute is a pretty soft paci and it is also very cheap so you won’t be out much by trying it. Otherwise, you might either want to just give it time to get used to it, or just don’t do it at all. You could try a teether toy, those are actually quite fun.
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Another thing to consider, is that pacifiers are actually intended to be not only a soother, but a workout for a babys tongue so they are better at nursing and also better at speech. So in reality, it is supposed to be a bit of work to suck on.
Try a smaller size, that could do it. I use a swedish brand (sorry for repeating myself like an old hag) for age 4-36 months, latex. I dont like the silicone, feels a bit plastic. And like Snivy says - it takes some time to get used to it. Sometimes I just hold it in my hand while going to sleep, that is nice too.
I find that while the silicone ones are great for driving and daytime use, I like going to bed with the latex ones; my wife says that i go through cycles during the night, at times I just sort of keep it in my mouth, and then all of a sudden I seem to suck on it furiously. I do know that I never have had a dry mouth problem; quite the opposite - I typically wake up to find that I have drooled all over my pillow or Mr. Tootles - my plushie bear.
I had the same problem as you at first, then it just kind of clicked after awhile. Its not so much in the lips its more of the the tongue pressing it up against your pallet. At least for me that feels natural.