Hi there and
to the forums!
It seems like you know what you're doing wrong and yet, you continue to do it!
You're moving too fast for your little guy-- two weeks is still a very short amount of time. When you first brought your budgie home, you should have given him two weeks to settle in, without trying to touch him or put your hand in his cage. You can do so now, to help him settle in and feel more at home.
During this time, be sure to talk to him often and read, sing, or talk quietly exclusively to him a few times per day. After the two weeks are up, you can put your hand in the cage as far away as possible and not move it until he is used to this; it may take days or weeks for him to be comfortable with this. After he is, move it a little bit closer until he's used to that, and so on, gradually building trust.
It sounds like he's slowly starting to settle in; the singing and chattering is a good sign. Don't worry about him playing with his toys, he'll start playing more as he becomes more comfortable in his new home.
Also, be sure to read through the forum's many Budgie Articles and "stickies" (threads "stuck" to the top of each subform for easy reference) to ensure you're up to date on all the best practices for budgie care.
If you have any questions after reading through everything, be sure to ask!