Oh do I remember those days. I swear, my son wanted to wear me like a coat that never came off. But they DO outgrow it, and the need DOES fluctuate a bit even before then.
I found that the more I fought it, the more he needed it. The best thing was to focus really strongly on him while I was with him, and then SCHEDULE TIME FOR MYSELF. LOTS of it, leaving him with a caregiver that I could trust and that he was comfortable with, and sticking to a REGULAR SCHEDULE for that.
We took it really slowly, introducing him to his nanny, with me staying with them for a few days. Then we developed a good-bye routine, and stayed firm with it. No changes like trying to visit him for lunch on a nanny day; I had to stay out of sight and out of mind. They had a lot of fun, really - she took him for long, long walks and explored all over town. They even made friends together.
I realize few can afford a nanny (I used the time to work, which pretty much only paid for the nanny, but ...), but a schedule can often be arranged with friends or family members. Make it part of the routine, and make it something that really allows you to center yourself and be ready to give the best of yourself when you are with your child.
These are precious years, but they are also really draining. Every parent goes through that, but even more so for parents with needy children like mine was.
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Mom to an amazing young adult AS son, plus an also amazing non-AS daughter. Most likely part of the "Broader Autism Phenotype" (some traits).