This will end, I promise.
We are still kind of going through it with dd who is 3 and a bit.
She is on movicol and we adjust it to make sure she goes every day. If she doesn't, I up the dose.
Like you, I thought she wasn't constipated as her poos were soft and she wasn't sore, but she 'leaked' poo all the time which I've since learnt is overflow. Sorry if tmi.
Toilet training definately helped as we could see what was going on rather than her just having a constant dirty nappy.
I wrapped a load of really tack cheap presents and put them in a basket in the bathroom. I told her they were poo presents and when she did one on the toilet she could pick a present.
I also got a copy of mr poo goes to pooland from a lovely mumsnetter, I don't remember her name, that helped with any worry about what poo is and where it goes.
I never thought I'd have the whole family standing round the toilet waving off mr poo and his children and a wee as we flush. Happy days!
Good luck and try not to stress about it.
It's taken a while and she's still on the movicol 9 months later, but it's worth it.