Hi K. -
Our 16 m/o son is like that sometimes. It was really bad a couple of months ago but he's gotten better about it over time.
I guess this is the time they really start experiencing social and separation anxiety. It was weird because he'd play by himself during the day, if I was watching him and nearby. But as soon as my attention was diverted, he'd complain and get all clingy. What worked best for me was this: I bought him a new toy - nothing expensive, just a cute simple toy. I also cleared out one of my lower cabinets in the kitchen, one that I didn't mind him playing in. When I would cook, I'd let him play with the new toy, but only then. That way, he began to associate the new toy with "playing alone" time. Then I started placing the toy in the cabinet instead of giving it to him while he watched me put the toy in the cabinet. That made his day because now he finally had a cabinet he could play in without getting yelled at! So he'd play with the cabinet, and the toy, and eventually he'd just sit on the floor nearby and play while I cooked. This eventually morphed into laundry time, where he would play as long as I was around. If I disappeared from sight, he'd just look around until he found me, then he was okay again and would just go play.
Nowadays, he may follow me around while I collect clothes for laundering, but he's not so underfoot that I can't do anything with him nearby. As far as cooking goes, unless he's really hungry he pretty much will play off to the side by his cabinet.
I should also mention that I'm a SAH mom too and we spend all our time together. There are no cousins near his age or anything, nor do any of my friends have kids his age. About three months ago we started letting him play in our church's nursery while we attended service. There are about eight other kids his age there, and I've seen a change in him since he started attending the nursery. He seems to be more confident and independent. That has helped greatly with his "alone time".
Hope this helps!