You're looking for unconscious competence. There are 4 stages of learning (according to this thinking), and you're at stage 3 - where you can do something, but you have to pay attention, which means you haven't REALLY learned it fully - from your description, that is precisely where you are. Once you get to stage 4, you can perform a task (in this case, playing a piece of music) without a lot of conscious effort - you just "know" it.
If you think about it, you may know a tune or two where "you got it" - you can play it without thinking. There are two reasons why that piece of music is at that state - either its easy, or you have played it some many times that you have achieved stage 4. Assuming that's the case, you know how to get where you need to get, it's just a matter of time.
How do you get there? LOTS of practice. I memorize everything for my gigs, and then polish the hell out of it - that's what gets you there. I do use other learning theory concepts to help - new tunes, if you practice them just before going to bed (plus a few other times in a day), your brain will consolidate that data better than if you didn't do that last run through in the evening, I also learn things in bursts - if I have 2 new songs, I'll run through one 3 or 4 times, then switch - too many times in a row, you get lost in the arrangement. So, tune 1 a few times, then tune 2, then back to 1, then on to 2 again, and they I'll do something else for a bit, and come back and do it all over again.
Doing this, I sometimes get into a real "flow state" when I play - it's almost out of body-ish in that you're there, but you're almost a spectator - I remember a couple times, playing and wondering how I fingered a certain passage, so I look down and just take it in as it happens - no self consciousness about it, my "fingers were playing", and I got to observe how I did something. One gig recently, I had a wave of goose bumps on one tune - maybe a minute where I was just having one goose bump/chill wave after another - the coolest feeling. I went back and watched the Youtube of that performance later - until I did that, I really couldn't tell you for certain afterwards if I was playing at that part or not - I was, and I did just fine.
The point of all this is your focus when playing should be on those around you - the other musicians, the audience. Music is a team communication sport - if you don't know what the other players are doing (you hear and see tings that help you understand that), you're not really part of the team. If you're unaware of what's happening in the audience, likewise, you're not seeing if what you're doing is getting through.
There's an old saying - amateurs practice until they get it right, pros practice until they can't get it wrong. Same concept, just different words.
en.wikipedia.org