I've been through a bit of this - widowed five years ago, 2 DCs. They did go through a fairly clingy stage and didn't like it when I went out, particularly one or two years on from when DH died. But at the time I was doing a course which meant going out for several hours one evening a week, so I persisted - I was lucky that at the time my mother was well enough to babysit for me, so they had someone familiar.
They have become much less clingy over time, and now are perfectly fine with me going out, and the older one is happy to look after the younger one (13 and 9). I don't go out that much, but I know I can when I want to. Longer periods off are harder - I've only had one or two nights at a time away from the children, and probably only half a dozen times altogether, just because it can be hard to organise childcare - they are still too young to be left by themselves overnight.
I agree that you definitely need a break and some time to do something just for yourself. If you are anything like me, the feeling of being totally responsible for everything, all the time, must be getting to you, so I understand your frustration with them.
Would you trust your DD to look after DS for an evening? I would suggest signing up for some kind of class or regular group - exercise, dance, book group, whatever - or just a regular evening with a friend, and making it part of your normal weekly timetable. And it sounds like you could do with maybe a weekend away - do you have any family they could stay with for a couple of nights?
They may kick up a fuss the first few times, but you do need to make them realise that you need a life as well. That may sound harsh, but in the long run it is not healthy for them to be in control of what you do with your time.
There is a temptation, I find, to think that since my DCs have lost a parent, everything else in life should be made easy for them, and they shouldn't ever be upset etc, but actually acting like that would do them no favours long-term - they would just grow up spoilt and self-centred.