Hi there. as a parent of two kids, I can tell you that this is pretty common and a sign that your little one is very attached to mommy, which is a GOOD thing. You'll find this time is fleeting . . . and before you know it, you'll be CHASING him as he tries to get away from you.
With both of my sons, I found that I could get small bursts of things done. At one, kids don't really entertain themselves much. That's just the nature of where they are at developmentally. But I could give them some magnets in front of the frig when I was trying to make dinner, or put them in their high chair with some little toys, pull out a bag of trains that they hadn't seen all day and they'd look through it. But honestly, developmentally, one year olds do very much want to be near mom and will either follow you or cry for attention. Normal, totally normal.
Now, I have two kids, my oldest has sensory integration disorder. This is a developmental delay of the nervous system. But it was actually my second son that was much more attached to me. Now at age 10 and 9, my formerly clingy younger child is his own being. He has no developmental issues or problems.
I believe in showering our kids with love. Houses can be messy for a couple of years, laundry can wait, etc. So, go with the flow and get your little breaks of freedom to do something that needs to be done and otherwise, you have a another year or so before real independence from you can be expected. I would never leave a one year old to just cry but rather would try to distract them with something else. Tough love is for a bit older for it to have any meaning based on my experience with mothering and children in general.
You know what I loved when my boys were that age (and when my first son was 1 years old, I had a second new born! Talk about a busy day!!)-- was a mother's helper. These are the tweens that aren't quite ready to babysit but can come to your house and PLAY with your child for an hour or so while you shower, do something that needs to be done, etc. The pay for that is usually 2 to 3 dollars. I had one a couple of times a week when my boys were little. They were great--- great way to get a little free time while the kiddos are entertained.
Again, totally normal and a phase that most kids grow right out of. good luck
Hi kirchoffha and welcome to the forum.
It is too early to jump to conclusions, but my suggestion is to consider
exercising "tough " love.
Paying NO attention to his fussiness and /or crying (as long as it is safe and nothing urgent is going on) for a while, till he gets used to your "non-response".
If there's a significant underlying reason for this, chances are, it will persist,
no matter what you do, right?
Best wishes to you and your son!
Niko