That must be really stressful for you, I really hope it gets better soon.
I'm wondering more if it is attention seeking rather than anxiety. Was he upset a few times and threw a fit when he wanted you closer and it worked,so he has decided it worked well for him and has just continued it?? I think at this age the kids test the water a lot to see what works and he loves you and wants your attention all the time, so the fits gets him what he wants.(he is a smart boy). Of course I have no idea what the situation really is, just a thought from my experience with my kids.
my suggestion is to give the hugs and kisses before you walk off to do something, tell him you will be back and tell him that he does not need to scream etc, and when he gets upset just ignore it(the best you can) do what you need (maybe something quick at first) and then come back like he never threw the fit and sit down and start playing cars or whatever he likes. If he chases after you, hold out your hand but don't say anything or calmly tell him you do not understand him when he screams(or whatever it is, however if he is screaming to much he won't hear you anyway and the hugs and kisses probably just encourage the negative behavior. but if he talks nicely you can give him the hug or whatever but you will still need to do what you need to do.
Or ask him to follow you while he hops like a rabbit, slither like a snake or anything to distract him from his negative behavior. Or everyone time you walk out of the room, maybe you can have a silly or funny noise you make to him, or he has to make. Especially in the home, you could ask him to play a animal game with you, by listening quietly for the animal noise you will make while you are heading into the other room, like a owl, cow, and see if he can make the same noise back to you. Basically anything that distracts him. Good luck, hope something helps.