I had necessary surgery for TOS. They found scar tissue wrapped around the thoracic bundle and scalene muscles. I still suffer 2 years later, I have nerve damage from the TOS. I have the same issue of the shoulder hanging being painful. One thing that really helps is having my shoulder taped up twice a week with KT tape. I have treatment from a neurological chiropractor and he tapes my shoulder. It is amazing how much the tape adds support. I have been wearing it for a year now. I probably looks funny, but the comfort I get is worth it.

Maybe something like this could help. I have them and do wear it sometimes when doing chores. I love it when I pull it very tight and pull the shoulder back as much as possible. http://www.amazon.com/Clavicle-Suppo...clavicle+brace

I also have similar challenges - dishwashing is very painful bent over the sink and my clumsy hands break everything so the family does most of that. I take clothes from the dryer and fold in a comfy position taking my time. Driving is horrific and I plan on recovery time after anything over a few miles (DC traffic is horrible).
I have found that a sling on my right arm helps me sometimes - especially with chores...I'm right handed so it helps to prevent overdoing it or recovering when too late. A soft c-collar also helps me in crazy traffic not to move my head too much...will wear around the house as well when trying to be productive.
I basically have a plan that physical effort of most types will mean downtime to recover and family understands now so that helps a lot.

Clavicle support
 

I did try something like this but I couldn't handle the pressure coming across the shoulder. I try to be careful with bra straps, sometimes they cause me pain.

I hired a chore provider through the League of Human Dignity. She comes three days a week and helps with basic chores like dishes, vacuuming, and laundry. It is $8.00 per hour, which is great considering I have been able to work more aka keep my health insurance.

I've had TOS for about 3 years and luckily am doing much better since doing PT and trigger point therapy. I am by no means cured. Unlike you, I can carry out basic chores. But, the more physical ones, like yard work, still set off my symptoms.

I used to be able to spend all day on projects around the house, only taking a break for lunch. Those days are long gone. I own a house with my sister and get very little support. Amazing how when you "look fine" people you thought were close to you will ignore your distress.

I tackled some bush trimming over the last few weeks and got no help from my sister at all. She only came outside to see what I had done, acted unimpressed, and snapped a photo of the bird nest I discovered while pruning an azalea shrub to post on her facebook page. My shoulders/arms were sore for days. I did some more trimming today and stopped after my arms went completely numb. She sat inside on the computer. After coming inside, she wanted to know if I wanted to go out to eat."After my arms stop being numb" I told her. She left the house without me.

Thinking it might be better to just hire someone to do the work next year, then make her pay half the bill. That way I can sit on my butt too and not be in so much pain for doing work that is completely unappreciated for how much effort it took me to complete.

Amazing to read these posts as I had/have the same problems. As I have beat TOS back bit by bit through conservative means, I have been able to increase the amount of time I spend on housework from around 5 mins to 40 mins, which is enough to keep my place livable.

One trick I have always used is to get familiar with my time limit and respect it. In other words, bail out before you're going to flare up. If 20 minutes flares you up then set a timer for 15 and stop. Every few months you can test your limits to see if they have progressed.

I found that wearing a tank top with a tight built in bra helps me be able to loosen bra straps so they don't hurt.