Here are some thoughts that might help.

I was going to suggest you try hide glue like OBG also It has a 30 minute open time and the wood will slide better. Claims have been made tha it gives better full coverage in the joint than PVA in very tight joints. Use it warmed up per directions and the joint will go together more easily.
I just warm it with in a bath of hot water out of the tap ( 125° F or so – 52°C). Lastly, it is reversible with heat and moisture. For a longer open time, warm the wood.
The downside is you have to keep your project in clamps longer.
If you can’t find OBG, Titebond’s hide glue works ok too, or make your own by salting hot hide glue. Plenty of info on the web.

But that doesn’t solve the issue of the wood swelling before you complete the glue up. Work quickly and it won’t let the water in the glue time to enter the wood with either type of glue.

If you want to stay with PVA, try Titebond 3. It has a longer open time than original or type 2 ( 8-10 min vs 3-5 min) Titebond says you can thin it 10% with little loss of strength.

And while a joint fit that won’t come apart even before glueing is impressive, it isn’t always a practical way to size your joints. With larger work. Consider a thou or two allowance for glue in the joint. It will still be plenty strong enough. Try wetting the tenon and then putting the joint together without glue to judge when the joint is too tight. Shaving the tenon is a great job for a shoulder plane. A router works also. Sandpaperglued to a board works in a pinch.