Some kids have sensory issues which affects their relationship with solids - so your comment that she is 'fastidious' and hates the texture of food makes me wonder about sensory processing issues (sensory processing disorder).
Give her one thing each meal that she has eaten in the past. Put a small amount of a new food nearby and see if she tolerates having it on her plate/tray. Your expectation is not for her to eat this new food, just to get used to it. It sometimes takes 10-15 exposures to a new food before a child will try it.
Choose something new with a texture similar to things that she's liked, or something with a distinct texture in case she does try it. Crackers or things with a crunch are often good starter foods because they don't surprise a child with the texture. Crackers also breakdown fairly quickly and so aren't much of a choking hazard.
Let her feed herself as much as possible. I'm a firm believer in the benign neglect school of feeding solids. Your job is to provide it, her job is to eat it. Our dd hated baby foods and so went straight to small pieces of what we were eating when she was about 9 months. I put it on her tray, she ate what she wanted to of it. Rice, peas, spaghetti noodles and small pieces of meat (smaller than a cheerio) were what she liked best.
Finally, there are a couple of books that I'd recommend:
Feeding with Love and Good Sense by Ellyn Satter (Ellen Sattyr?)Just Take a Bite
Just Take a Bite is for kids with feeding/food aversions, while the first is a more general book on feeding kids.
If you think that your daughter might have other sensory issues, you might want to read: Sensational Kids to see if this issue with food is part of a bigger picture or just her little quirk.