Man, lythium ninjaing me with her super simple explanation that's much better than mine.
Listen to her.
Well, you're not really supposed to punish. That's why it's considered a disadvantaged position. The best decision is usually just to block.
If the opponent is being fancy and just spamming overheads or something then you can get away with jabbing them out, but I rarely got to do anything like that because you always just tried to jab immediately so I just went with the simple 2a/5a spam oki which stuffs mashing on wakeup pretty easily, lol.
Things like DPs, backdashes, rolls, etc can work occasionally as well but if you get too reliant on them people will see them coming and hit you out of them. Rolling out of the corner is especially risky because people will be looking for it.
You were waiting quite awhile in the swallow moon before you did an attack out of it, and usually I could react to it and try to jab you out. It's just something that comes with practice, I guess. You can hold up for a bit which makes her float higher and then try to attack, which would make it come out faster than just waiting there. You can do j.a immediately after a TK swallow moon and it'll come out regardless of your height.
Also, when you get an item, you usually just use it over and over until it's gone. Like when you had cats, I could just sit there and block until you ran out (I tried to barrier cancel dash between them and I ended up getting hit out of it most of the time because i keep forgetting that doesn't work as well online). Same thing with the hammers, which is a worse idea because it's like -9 on shield plus the startup of the next hammer. That's plenty of time for me to attack you out of.