You say that you feel embarrassed to approach your boyfriend for sex. The word embarrassed is very telling. Many women feel scared to be rejected, but you said embarrassed. To figure out why initiating sex embarrasses you, you need to ask yourself some questions. First, why am I embarrassed to be the sexual aggressor? Do I feel ashamed to admit my sexual urges? Do I feel more comfortable when my boyfriend asks for sex because I can pretend that he wanted it and that I am just giving in?
Many women who feel ashamed of their sexual urges prefer being asked for sex in the middle of the night, for example, which allows them to pretend that they were half asleep and didn't know what hit them. I think that your own questioning will reveal that you aren't fully comfortable with your sexuality.
To not initiate is a way of sitting on (no pun intended) your own sexual urges. In not approaching, you can deny that you even want sex. Then, when he approaches you, you can also pretend that you are only giving in to please him.
Obviously, it would be healthier for you to own your own sexual urges and feel good enough about them to ask that they be satisfied. To ask for sex is to be entitled to your sexual desires.
You aren't there yet, but you sure are asking the right questions. Ask and ye shall receive in more ways than one!