Go with your little one and be around people who love you and can help you.
I am a single mum in the UK and get very little help. Apart from tiring, it can be lonely too. You need adult company as well.
In the UK if you earn under a certain amount, (£16,000) your child is entitled to 15 hours a week nursery care from the government. Are you taking advantage of this? That could be 5 mornings for 3 hours or five afternoon sessions for 3 hours. Some nurseries let you do it in other ways too. So maybe over 2 days?
Its private daycare nurseries too. Get him signed up he will have loads of fun, eat great food, make friends and become more independent.
Personally I think your partner needs to realise that life does not revolve her and you visiting her.
I think if she see's you establishing a good life and routine for yourself and your son, it might make her work a bit harder at being sober. Let her see that if she continues on this path she will miss out. Don't pander to her every need and want.
Sure start centres in the UK also offer help - you can take your boy to free play sessions and stay with him. They do stuff for Dads too like support groups for men doing the majority of the parenting.
You sound like a strong man who is realistic. You sound sensible and considerate too.
However be more considerate of your boy and yourself and less considerate of her.
You will get through this and you will be fine. That is said from my own personal experience and with love and support.
I wish you the best xx