Argh I remember this so well (my youngest is only 2.5 so it's not too far away for me). You poor thing OP it's horrible.
My eldest didn't sleep anywhere but on me in the day and woke - a normal amount - until she was 9m old. Then I put her in her own room and she has slept beautifully since, bar the usual teething/illness/new baby coming interruptions.
Second child slept brilliantly from twelve weeks and I thought I'd got lucky this time, then she hit eight months and my god the gates of hell opened. She woke every two hours, for hours, every single night until she was around eighteen months. I tried everything, except crying it out. Her room was the perfect level of darkness, she was the right temperature, she ate well, she had a great routine (and went to bed happily and easily). She just wouldn't stay asleep or go back to sleep!
Both of my children were mix fed, breast initially, then breast and formula and eventually formula only. Both weaned and ate normally. I never co slept, in our room they both had a Next to Me crib, so they were right next to me but not in my bed. Both moved to their own rooms at 9m.
In the end, after I'd spent a night in tears and then DH had had to pull over in a lay-by for a nap on his way into work, we decided enough was enough with my youngest and I got a sleep consultant in. Best money I ever spent. She rejigged our routine timings wise, we moved feeds, meals, everything over a matter of weeks (so it wasn't an instant fix) and tried a few different things including not getting her out of bed when she woke (we had been taking her downstairs to settle so she didn't wake the eldest). It took some time, she was never left alone or distressed, but she did learn that when she woke she needed to settle back down.
I'd say after about a month or so we cracked it. She's 2.5 now and we've had a year now of her going to bed at 7, sleeping 12 hours and still napping for two hours in the afternoon. Hopefully she's past the baby stage now so we shouldn't ever (I hope and pray!) go back to night wakings like it.
Your baby is not too much younger than mine was when we started, at over a year some gentle sleep training is definitely ok and possible. As I said I've never done crying it out methods I don't like it, but sleep training doesn't have to mean leaving your baby to scream. I do think gradually moving your child into their own room would be helpful though, the first thing to eliminate is disturbances and for all you know sleeping together could be waking the baby. Start with day naps in their own cot first, get them used to the move, the room etc.