I sympathise OP.
My eldest DS went on a gap year to stay with his DD in South-East Asia when he was 18. I was used to my 2 DS's going out to spend holidays with him but this time he went out of sight at customer I was distraught. I felt like I'd lost him, and in many ways I had. I also had regrets about how I'd brought him up and all sorts that I knew I could now never put right.
In many ways he left here a boy, and came back a man though. After 6 weeks back at home he went to Uni which, at 100 miles away, was much easier but I~ still missed him. I had far from an 'empty nest' with 5 DC's till at home but regardless, it's was a wrench, and I remember feeling very depressed after he went.
My second DS going was not as bad as it was only 30 miles away, but still, it felt like a massive loss. It's huge thing, and I also found it triggered a massive sense of abandonment, which I realise is illogical but it's still there.
I'm not sure if I'll ever really 'get over' having to say goodbye really - I make a point of not watching either DS's walk off when I drop them somewhere- I find it too upsetting. I broke this rule a while back by watching DS 1's ship leave for a 6 month tour (he's 27 and in the forces now) - I stilled cried my bloody eyes out!
I suppose it's different for everyone, but to me - DC's leaving is a life changing thing that you have to learn to live with, but you have to smile, wave and let them go. Suffocate them and you'll ensure they keep their distance, let them go and they'll come back when their ready. They'll also one day respect you one day for giving them that freedom with no guilt attached.
You'll find your way OP but for now, be kind to yourself. It's a difficult time. 