Your trap may not be the problem , especially after the clean out. It may be other parts of the drain system that are either split or rusted out where you can't easily see or simply not tightly secured to each other. If you and your SO can dry things out, light the system up front and back for good visibility, and then one of you pour in a pan of water while the other watches, you two should be able to pinpoint which spot or spots the water is coming out from.
In the first place I ever rented, the pipes under the kitchen sink would overflow/flood ONLY when large quantities of water were sent down the drain. Someone did a botch job on the plumbing which my Dad spotted when he came to help me sort things out. The straight pipe coming down from the single sink into the trap was not a single pipe secured into the trap. It was two pipes with nothing actually joining them and the pipe from the sink simply rested inside a straight bit of pipe that was secured into the top of the trap. The two were snug enough and there was enough overlap that they could handle small amounts of water. But fill the sink up as when doing dishes and let it drain, the water in the drain weighed enough to cause the lower part of the system to sag which pulled the two pipes apart and caused a flood under the sink. Dad fixed it enough to live with by using some metal strapping to add support to the drain pipe so it wouldn't sag enough to pull the two bits apart. If it were my actual house, I'd have fixed the piss poor plumbing job AND added the support to the drain to keep it from straining the system under a full load.
The problem could be right below the sink and the water just pouring down onto the trap then to the cabinet floor, making you think it's coming out of the trap. I recently learned from an "Ask This Old House" episode (a US TV show that helps people fix issues around their house) that the strainer in the sink is made up a nested set of bowls or baskets and that the lower one under the sink can rust away and let water leaking out of the sink make it into the cabinet.
https://www.pinterest.com/pin/449374869047978699/ has some good illustrations so you can see how that all fits together and why, if it fails, it'd make a mess.
You may just need a new gasket in a place or two or maybe some plumbers tape in joints to get a better seal. Or you may end up needing to replace a few bits of pipe. If you post a picture or two and can point out exactly where in the plumbing water is coming from, people here can give you really specific advice on what needs fixing.
