I don't have kids in this situation, but I am a classroom teacher who frequently finds herself like this. This is what I would do based on classroom experience:
Set out clear expectations before the behavior starts. Is it a time that it is okay that they get loud and wild, or are you expecting quiet(er) play? And carefully assess how often you really expect quiet play, even at dinner.
Maybe one dinner a week can be "crazy" dinner, where manners get tossed out the window. (You and your husband can eat later and the girls could all eat together). Then, when behavior other nights gets out of control, you can tell them to save it for their "special girls dinner."
Make sure that you are actually punishing violations of rules, not just annoyances to you. If there is a "no pushing" rule, and they start pushing, it's time for a time out. If they are "raspberrying" each other and you just think its a little gross, let it go. They're kids and they're entertaining each other.
Give one clear warning for a rule infraction, specify what the punishment is going to be, then follow through if the rule is broken again.
Have the most important rule in the house be "you stop and listen to me when I am talking to you." Then, make sure you aren't always interrupting to punish. They're going to be more likely to follow the rule if sometimes you interrupt to say good things. (I'd be more likely to stop playing for "let's all go out for ice cream!" then "I said stop that! Into time out for you!")
Finally, I would institute some quiet or separate playtime every day. It's possible they really don't know how to play independently. Even if it's just 15 minutes. I would frame it in the positive, and when playtime is over you could all meet and report back about what you did. 15 minutes of quiet goes a long way.