Hi! Social introvert here. Social because I do tend to get along well with people and enjoy company. Introvert because if I do social things too long or too often or with too many people I don't know, I either fall over exhausted or start biting heads off. So, here's my advice...
-Don't try to be an extrovert. Your preferred form of socializing is probably fewer people, more people you know, and a quieter environment. Coffee or lunch or game night rather than bar or house party or concert. Drinks can still be involved, of course, but you will probably be happier with something more low-key.
-It's ok to be tired. You might feel the next day like you went on a 20 mile hike, uphill both ways, when you just got coffee. Like someone mentioned above, it means you're working your "extroversion muscles." It's a good and normal thing. Embrace it as a sign you did well. [Corollary: if you need to step away from a group event for 10 minutes and take some quiet time, that's ok, too.]
-There are more ways to strengthen connections than always spending time together. Be a nice person. Bring treats or small gifts for acquaintances and coworkers for birthdays or holidays. Offer to babysit or help out with a lift to the airport. Like stressful experiences, these small kindnesses bring people together.
-This is hard to do on purpose, but try to find an extrovert you can 'leech' from. By way of explanation, my bf is an extrovert. I find that (a) his need to socialize tends to get me out and about, and (b) that socializing when he's around doesn't wear me out half as much; I'll even volunteer to go to crowded, loud events with him (I still have to be a recluse the next day, but I can get through it and enjoy it). Obviously OP and others already have introverted partners, but I also have friends of a similar nature--they are exuberant and make most of the effort in socializing, making it easy for me. And they make encounters with other people easy if they're around.
-Practice small-talk on strangers. Your cashier at the grocery store or a stranger at the bus stop. Compliment their clothes/hair/jewelery, talk about the weather, local or national news, anything. Smile. If its awkward, you'll be out of there in 2 mins anyway, and they probably won't remember you the next time you shop there (or whatever). [Not that you should force conversation, but remain open to it, unlike how I used to be where I felt any interaction besides "Credit....Thank you." was excessive.]
-Practice conversations in your head/with yourself. I know, this is weird. But when I knew I had to have a conversation with a stranger (phone conversations were the worst), I would run through what could possibly be said while doing other things. Helped me feel prepared with phrases/responses. Now I only do this for interviews and such.
I'd also like to second making rituals and inviting people to your house. Just not too many at once. More than 6 tends to be a crowd. :) Although hosting a crowd tends to at least give you something to do/focus on so that you don't feel so pressed to socialize. On the other hand, you don't actually socialize that much.