Unfortunately you do need to be able to do both if you plan on swimming laps for recreation or competitively.

I don't know of any regimented training program that teaches this to adults but I can help you try a few things.

You need to start out slow and force yourself to float on the water and then turn to the side to take a breath.  You need to practice this over and over again until you feel comfortable.  When you do then put movement into it.  You will need to take that breath and also take a couple strokes.  When you can finally do this successfully then you will have to work full strokes into this.  I would set a goal of swimming one length of the pool and see if you can manage that.

There are common sides for people to breath on but there is no correct way.  I happen to breathe primarilly to my left which is less common than to the right.  I can breath to the right but it does feel more awkward.  See, I know how it feels.  But remember, it was awkward also for us when we breathing to our accustomed side.  You just need to force yourself to do it.

For body position, you need to make sure that you are finding a way to float and then lift your head out of the water.  Make sure that you are at first just getting it out and then after that you can work on hydro-dynamics where you are just getting it out enough to take that breath.

Please work on these things and after a while I think you will get it