I think the jacket is working exactly as designed. I don’t think it ever had any water resistance so anything you get is going to be something you add. A dry cleaner doesn’t have a waterproofing that is going to seal the jacket fabric; something like that would effectively be coating the jacket in plastic which would change the characteristics of the jacket completely. Cotton/nylon is porous. Real waterproof jackets are either solid plastic or have a layer of Gore-Tex (or equivalent) that is made with a semi-permeable membrane that allows water vapor out but doesn’t allow water droplets in. You can’t add that aftermarket.
So, the best you can do is make the water bead up on the outside of the jacket. That’s what’s on the outside of Gore-Tex jackets to make the water bead up, for for the same reason. It prevents water from being absorbed in the fabric, at least for a while. It will overwhelm your cotton/nylon jacket eventually, but it’s probably the best you can do.
There are silicone waterproofing sprays that are generally used on boots, tents, and tarps, and the like. You could try it on your jacket, but try it on a small section first. It may change the characteristics of the jacket so much that it’s not worth it. I’m sure a dry cleaner could do it, but there’s no reason not to do so on your own.