Nice to finally meet you!
There are some great posts already. Volunteering, matching sites, and being the designated driver all sounds like great things for you to try and do socially.
I can definitely relate to the same shit, different day routine. The thing that helped me at home and getting over the SSDD routine, was by simply trying new things. These things don't need to be spontaneous, expensive, or something you need a group of people for either (which is a plus if you're introverted like myself). And you might just find a new favorite hobby too.
I started reading, which is something that I never really did. I always loved arts and crafts and by doing those I felt really accomplished. I play guitar a lot, learning new songs and just taking care of my guitar is a nice change up from the everyday 'work, home, dogs, eat, wash, social media, bed'. Just by adding one new thing a day can really change your mood. That really helped me. Even if you think you might not like certain hobbies, you need to give them a chance if the opportunity shows itself or you think you might be interested in something. I know plenty of people, myself included, never wanted to do/play a certain thing before and now it's their favorite thing.
As far as the gym goes... I can't be confined to a few rooms of exercise equipment. I need to be out riding a bike, bowling... hell if I had money i'd probably be doing some airsoft.
I don't know if any of these things will help you, but just know that I read every word in your intro and every word you say means something. So spill your guts to us, we will always be here to listen. No one should go without not having anyone to confide in.