My recently-acquired previously-neglected mare started out very headshy due to a lot of tenderness in her ears. I've modified a lot of typical handling to be very careful of her ears: if I can, I don't even halter her; if I do halter her, I take it very slowly and make sure to bring the crownpiece around well behind her ears; when working on her face or neck near her ears, I always start well away from the ears (halfway down the neck, or closer to the nose) and work my way up.
If you can figure out just what it is that sets this filly off, you can do something similar. If it's her ears that she's touchy about, avoid the ears while you're developing your relationship. If it's her eyes, stay away from the eyes--or forehead, or nostrils, or whatever. Ears are probably most likely for a touchiness due to pain of one kind or another, but a horse who's had bad experiences may have negative associations with any area.
I wonder if her reluctance to be caught is a result of the head-shy-ness--that is, when she's caught, she's haltered; she dislikes being haltered; being caught is therefore unpleasant.
You can try using a neck rope instead of a halter for a while. Don't put a halter on her at all, just loop the rope around her neck (most control up toward the jaw, but much more gentle toward the shoulders) and lead her with the rope.
You can also try spending multiple sessions just catching her and letting her go, *without* haltering her or putting a rope on her. The goal of this is to help her see that being caught can be pleasant.
I've got a few more tips on catching a horse here: