First, welcome! I'm also very new -- picked up a tenpin ball in earnest less than 6 months ago after half a lifetime in a different type of bowling not found anywhere in the world outside of New England (where I grew up) and maritime Canada.
I bowl with an 11 house ball, but I have to choose between holes that are too big, or too small and I use the holes that are slightly too big.
Here's your problem in one sentence. If you want to assure yourself frequent, pain-free bowling, you're going to need your own ball fitted for your hand. If there are any people you know around who are really avid players w/a few balls (or more than a few) in their arsenal, ask them all where they get their gear and where they get it drilled.
It's a big deal
to ask where to have your ball(s) drilled. The finger pain that I've had came from a drilling of my first ball that the nearest Pro Shop
botched 6 ways from Sunday. My ring and middle fingers would throb in pain for 3 days after every session. As you might guess, I won't let them touch anything of mine again and have switched to a different shop who plugged the offending fingerholes and re-drilled.
(My middle finger is a little tender now, too, bc I made the mistake of tossing off 12+ games in one night 2 weeks ago. The scorer was off and I was practicing, so I don't even know exactly how many it was. I didn't feel any pain that night, but the next morning I only had about half the normal range of motion in the knuckle closest to the fingertip. It only rested itself back to normal yesterday.)
It's probably going to kill you to hear this, but STOP BOWLING until the swelling, tenderness and pain go away. If you hit the lanes earlier than that, you'll only make it worse. It doesn't sound like you bowl in any leagues at this point, so there's no worry about missing anything or putting teammates in a difficult spot if you're hurt.
It's also worth noting that if you're using equipment that fits your hand properly, you'll be able to use heavier gear than 11 pounds -- 13 at a minimum would be my guess. In the month I used house balls, 13 seemed to be the best weight for me to handle. 14 would feel fine for a game or two but after 3 or 4 it would wear away at me and feel like a bit too much. 15-pound gear drilled for my hand works just fine.
I'm just starting to learn to generate some hook, and there are some very knowledgeable people here who have tossed little tidbits my way to help w/that. Please keep asking if anything's on your mind!