Firstly, huge hugs for you Tammy

. I'm so sorry you're having such problems organising a special trip for your family. I know it may not feel like it now, but you guys will have a magical holiday, whatever you end up doing.
I'm sorry I can't offer a huge amount of help, but I'll give you what I can. Keep chasing your contact until you get what you need. Work out exactly what it is you want from them, and stick to your guns. If you just want any nice cake with a candle in it, at a certain restaurant or location, tell them that; if you want a particular flavour, design or size, let them know what you're willing to be flexible about (e.g. location). I'm sure you've already done this, and I don't mean to suggest you don't know what to do (you sound like you're handling them pretty well already!), you just have to make sure they know what you will and will not compromise on.
If they have problems with something, or are too vague, ask if you can email one of the chefs to discuss it directly with them. Disney do try to do right by their Guests, but unfortunately they're a big company and things do get lost in the pipelines (I'm not saying this is an excuse for all the trouble you've been through, of course).
If anyone has any special dietary needs, talk to Brenda Bennett about special cakes.
As to not getting a response on the TTY, this isn't acceptable, and isn't very 'Disney'. Write a letter of complaint to them, outlining exactly what problems you've had, how long you've been trying to arrange this, and how you feel they can fix things (getting people to answer their textphones, would help for a start!). I've heard very good reports of Disney's complaints' department, so hopefully you'll get a satisfactory resolution.
I don't know if this has been discussed on another thread, so I'll just be brief. Your DF might find touring the parks easier if you rent an
ECV, so he can speed around with you all day. To take some of the stress off your back,
you could talk to Guest Relations at the first park you go to about getting a Guest Assistance Card, to allow you to use your DD's stroller as a wheelchair. This will let you take it into lines and pavillions, so you don't have to carry her. I've heard mixed reviews on how easy it is to get a GAC for this, but if you have any problems just ask to speak to a supervisor. Information on both
ECV rentals and GACs can be found in the FAQ section, if you haven't considered these yet.
I hope none of this sounds patronising; I most certainly don't want to suggest that you don't know what you're doing! I just know how easy it is to get locked into these things, especially when you're so emotionally invested in the results, and sometimes a fresh pair of eyes can push you in another direction.
Good luck, and I hope you have a truly wonderful time!