In the gentlist way possible, it sounds like , for a very long time, you've taken the easiest options with regards to sleep and now that's come back to bite you in the bum.
Instead of doing what's easy, you now have to do some hard work with some parenting that's tough to do, rather than easy.
So gear yourself up for the fact this is going to be hard work and before you begin ask yourself - do you have the resilience and resolve within yourself to cope with the hard work this will take?
Taking the easy option isn't a problem if you recognise you're doing it and know you won't get great outcomes as a result. But it's a really bad idea to begin trying to solve this and then give up (as you have with previous attempts to solve this).
So, what's important for babies/children of all ages is to go to sleep where they stay asleep. This is the central pillars in all aspects of good sleep hygiene. Moving an already asleep child will never give you good sleep habits, as you've found.
At aged 3, there is absolutely not other options - getting your toddler to sleep in their own bed and stay there is going to involve a lot if crying. A massive, huge, monumental amount of screaming. Are you ready for that?
The solution is the definition of "easier said than done" but is very simply just to be consistent, repetitive and don't give in.
You put your toddler in bed, say a mantra that defines what you expect ("sleep time now, we lie quietly in bed at sleep time, nan night" for example), kiss and leave.
Child gets up, or screams. You just act like a broken record. Return, put in bed, say mantra, kiss and leave. Repeat, repeat, repeat, repeat, repeat, repeat... This will take a longggggggg time.
You do this same thing st every wake up. Every time, every night, always the same. Expect a really hard 3 weeks or so. If you are consistent, you will can solve this.
In other contexts this problem is caused by not having firm boundaries for your child. Toddlers need really firm understanding of behaviour that is acceptable and behaviour that is not acceptable. And your responce to unacceptable behaviour has to be consistent so they know there is zero wriggle room when it comes to unacceptable behaviour. This applies to all situations, not just sleep.