Agree completely with Amoafwl. Your "expectation" that both of you will keep the house clean is not going to cut it. I would not do anything for her, but you do have to do things for your kids. Your kids cannot live in filth, so I would maintain the common areas, and yours and their rooms. Do not clean her room. Do not do her laundry. If you have a common hamper leave her clothes in it, take yours and the kids out to clean them.
Same with the finances. You can ask her to pay her share, but if she doesn't, what are you going to do? Jump up and down and scream? Hold your breath? Have a plan here. Buy only the necessities for you and the kids.
Where are here accusations of dating coming from? Likely two places.
First if there is an OM then yes it is projecting. "If I am dating then he must be too." This happens a lot. I think all of the signs of an EA and/or a PA are there in your sitch. However, it doesn't matter. You handled it pretty well, keep getting better at it.
However, it may be that she is feeling the loss of control over you. Most WASs do not handle that well. They will almost always try to find a way to reestablish control. After the LBS has begged, cried, pleaded, used logic and reason, pursued and pressured for a period of time, and then, POOF, that goes away, the WAS is all "what the heck happened? Where is he? What is he doing? Why isn't he bugging me?" It is classic pursuit-distance. Look up that thread, it is always worth a reread.
M(53), W(54),D(19) M-23, T-25 Bomb Drop - Dec.23, 2017 Ring and Piecing since March 2018