Hi - first welcome to our community :-)
I am sorry to hear that you have recently been diagnosed with PCOS. Did your Dr give you much information on it? If not, below is a link that you may find helpful: www.mayoclinic.com/health/polycystic-ovary
As far as exercise is concerned, you don't really NEED to have designated exercise time as in a gym or an hour or so. You can get many little snippets of it in during the day. For example, at lunch time you could go for a 10 minute walk; maybe walk up and down the stairs 2-3 times. Where it comes to your laundry, you could put it away one at a time like I do. That is actually the bulk of my exercise (I love doing laundry so that's no problem:-) and putting my groceries away one at a time. When you vacuum you could put some music on and dance while you are doing it. If you watch TV, have you considered doing some chair exercises? There are lots of them on SP in the exercise section. Get yourself a gym ball and sit on that while watching. It is really tempting to bounce up and down on it and end up rolling around the floor. That all helps. You can sit on it while on the computer. That takes energy just doing that because you need to maintain your balance. If you DO want to use the gym membership, would your husband go with you on a date? You can both chat while on the treadmill.
Make sure that the groceries you get are healthy - fruit/veges, healthy protein and nuts/grains.
Where it comes to your being short on time for doing things, are you trying to do too much? Make a list of all that you do, and then go through and cross off the things that aren't important. If, for example, you vacuum a couple times a week, you can reduce that to once a week or little longer. If you clean windows every couple weeks, what would happen if you did them once a month? I'm sure you can add a lot of things to that. It is a case of learning successful time management skills.
Good luck and take care,