I have found the same thing in my past relationships; I consider my self to be fairly sexually dysfunctional for a few reasons, but sex is a lot more enjoyable, and you want it more, when you feel emotionally close. To achieve that closeness, it is important to spend intimate time together that *doesn't* involve sex on a regular basis. You're right, this does tend to be more important for women than for men. So the best thing you could do for now is to tell him exactly what your needs are. Is it that you are not making enough time for each other? Or the opposite - do you see so much of each other that things feel monotonous for one or both of you? Taking time away from each other can make time together more special. I'm sure you have heard the advice before, but especially since you have kids (I don't know how old they are so this may not apply) try to take one night a week to get a sitter and do something you both enjoy, which might mean sending the kids out for a night and being together at home.
The other part of intimacy that you mentioned is the trust issue. I don't know your background so I can't comment on it, but you might explore what exactly the lack of trust stems from. Does it come from a lack of openness between you? Did it start with a past betrayal? Is it as simple as feeling a lack of closeness?
I think in asking the questions you did in your post, you already have part of your answer. You know what you need to improve the situation (emotional intimacy), and that will require work on his part, as well as yours. He is being open and expressing his need for sexual attention. Maybe he doesn't realize that he isn't getting the sexual side of intimacy from you in part because you are not getting the emotional and non-sexual affectionate side of things from him. If your husband needs reassurance that you love him, give him that in whatever form you can. But he needs to respect your feelings as well, and show you in non-sexual ways that he loves you.
I hate the word "frigid", how it can make you feel like less of a woman or inadequate as a partner, and I understand how bad it can feel to not want to be sexual with a partner you love in other ways. I sometimes forget what it feels like to be a sexual being, and how important that is to others, just because it doesn't feel important to me at the time. Medication can be a killer to intimacy, with one partner withdrawing when needs are not met, and not meeting the needs of the other partner, who in turn feels less inclined to meet the needs of the other... it snowballs. Don't let it go on. Even if the sex part can't be solved right now, the intimacy part can be. Be open and ask him to work on a solution with you and everything else will fall into place.
For the sake of your relationship, is it possible to consider a different antidepressant? Have you tried Wellbutrin (which is sometimes prescribed for ADD as well) or Trazodone? They both have really low instance of sexual side effects, and in some cases they increase sex drive. I don't know how many meds you have tried, and which work for you and which don't, but it might be something to consider. When a medication improves one area of your life, but decreases the quality of other areas of your life, you sometimes have to make a decision on whether that particular drug is worth it. I can't answer that for you, I'm just saying I know how it feels because I've been through it.
Have you talked to your doctor about your loss of libido, and the idea that it might be caused by your cholesterol drug as well? I don't know much about cholesterol lowering drugs except that certain drugs (I don't know which one you are on) can lower your body's hormone levels. Is there anything else that the doctor has recommended to lower your cholesterol, such as change of diet? Is this medication a long-term solution, or a short-term one to get to target cholesterol levels before maintaining them with diet? If that is the case, there is time to work on the intimacy issues and be able to look forward to the time when you won't have to use that drug any more.
I hope you find a solution. In the mean time, don't beat your self up. You're not the first woman to go through this, you are not alone.
http://www.webmd.com/sexual-conditio...-drive-killers
I found that slideshow affirming and reassuring, maybe you will, too.
Good luck!