Something’s not right
Welcome to SANE @glikescat I expect there are many others here who will offer support for you in this awful and difficult time.
My experience of pregnancy, some 40+ years ago, was similar, I vomited multiple times daily from the day of conception to delivery. I carried a bucket with me. I did not know pregnancy could be like that. Now they know it is a particular condition that needs a particular treatment (can't recall the details) but back then it was just 'really bad morning sickness' that lasted all day.
For the first one I took their medication, not that it made much difference, then paid the price by having a child affected in utero by that medication. (it's use has been discontinued and the class action lawsuit is finalised).
I understand things are better now and medicos are more diligent with testing and so forth before prescribing anti nausea meds during pregnancy.
I got through that and subsequent pregnancies using Raspberry Leaf Tea, the dry biscuit before getting upright from bed, tiny portions of food and water, and avoiding odours. I had to cook meals, but rarely could eat anything worthwhile be it at home or when out. Seafood (I lived inthe outback, so that didn't happen often), soup, and Arnotts chocolate cream biscuits were my standbys, and tinned beetroot vinegar. 
I'm sorry you feel like you are a burden to anyone, especially your family, that does not seem fair. You have a small child to care for and your husband is part of all that has transpired. Health is not something we have control over, and pregnancy is one of the most unpredictable experiences I think humans go through.
For the longest time I felt like everyone else had let me down by telling women that pregnancy and being a mother are the totally best, happiest and most fulfilling things that a woman can do.
We are mislead by the focus on the fairy story and the avoidance of the awful parts that get glossed over, if mentioned at all. Pregnancy is hard, really hard! Being a parent is even harder, and it goes on for longer. 
Do you know what being a wife and mother means to you, and is that a realistic outlook, or has it been overly influenced by the rosey idealism that we have been told is how it should be? Can you shift your expectations to be more in line with your currrent situation perhaps? Can you talk to your partner about how you feel and let him help you come to terms with this without feeling like it's not okay to need his support? He may have some neat ideas to support you that has not been thought of yet.
I don't mean to upset you, but I expect I probably will. I have asked myself all these things and more as my expectations crumbled around me.
Please talk to your care givers, your doctor, about how you are feeling. The medication might be able to be changed to help you function better.
Maybe you can get someone to help with the work so you can spend time with your little one. The housework will wait, but I recognise there are some things that need to be done in order for a household to function effectively. Is there a mother's group near you that can set up a roster for home help or something similar?
I do hope you find something to help you get through this comfortably.
@glikescat How wonderful that you have such a supportive environment, that would take some of the physical 'burden' of things like housework away for you.
I understand the need to feel like you are pulling your weight, and feeliing like you are sharing the load. The equality of a marriage, or any relationship, isn't just about the short term, which, in effect, a pregnancy is, it's about the long term give and take that ebbs and flows within that relationship.
You sound very level headed, would it help for you to feel less reliant if you were able to see the picture in a more long term sense?
The vomiting might pass, you're near the 3 month mark and often that seems to be the magical marker for the sickness to either lessen or leave. (although I can tell you that doesn't always happen 
) but it might (fingers and toes crossed).
At one stage I put Vick's or something strong smelling like cloves,(be careful of essentials with pregnancy of course) that I could deal with, under my nose (an old trick from a previous profession) so I could cook - maybe there is something that will work for you to at least let you feed bubby. A flu mask might help with changing nappies every now and again. 
Heh, I had to be so careful going out, had to scout the area so as not to pass by a hairdresser as the smell of hair dressing chemical brought on an instant throw up. Awesome look in the middle of the street!
Over time, I'm sure your current sense of dependancy on others for assistance will balance out when you support your husband through something difficult, if you haven't already. Your other family members are probably delighted to be spending time with your LO under the guise of helping you, and also feeling part of the life of the new one coming.
Try to be kind to yourself, which is soooo hard when you feel so nauseated. There is nothing, nothing, nothing, in my reasonably extensive experience, that is as debilitating as the depth and length and breadth of the disabling effect of extended, overwhelming morning sickness.
It will pass. 
