Perhaps you could get another cage for them to sleep in placed where they would not be bothered by everyone. I don't think that just covering them will work.
Budgies are a bit more skittish by nature to begin with, but they would all benefit greatly from a good nights sleep as well as time to fly. The more time that you spend talking to them and interacting with them one on one the more that they will trust you. The more that they trust you the more likely that they will step up for you and want to be with and on you.
Especially, due to the fact that the issue that you are wanting to address is a trust issue and is one of the basics to developing a good foundation for a developing bond flooding is the worst thing you could consider doing.
Although, I am fascinated by my birds now, when Kiki first came to live with us I did not want a bird in the house ever. The only reason that I took her in was that she could not survive the temperature of the season for another night as it was still spring her in the mountains and she is a tropical bird. She had some major issues, which I soon learned about the hard way, and being able to trust was a big one with her due to the lack of proper care she received from prior owners. In fact every bird that I have today was abused in one way or another, but I have never had to teach one of them to step up for me, as soon as they realized that they could trust me, I became their favorite place to be. They spend the day going in and out of their cages at will but when it is time to go to bed I must put each of them into their cage because they would rather go to sleep perched on me. Even though they will chase each other off of me during the day they will all perch on me to go to sleep at night if they can. Try this link and the training contained in it for your birds, [
viewtopic.php?f=11&t=227 ] .