Furniture, Wood and Cabinetry Finishing - Need help with refinished table

I'm refinishing a dining table and chairs. Here's what I've done so far and what problem I'm having with the table...

Sanded all existing paint off (down to bare wood)
Applied a water-based tinted primer
Brushed on 2 coats of water-based satin black paint
Applied a coat of clear satin oil-based Polyurethane by MinWax
Sanded lightly with very fine sandpaper and used a tack cloth
Applied 2nd coat of Poly
Sanded lightly
Applied 3rd coat.

My problems are that I have brush strokes on the table (even though I used a Purdy brush and feel confident that I applied a consistent level of product and gave ample drying time) and now, after the 3rd coat, the table looks smudged/foggy.

What do I do now? Do I have to sand the darned thing all the way down and re-do everything?

Thanks for your help!

Repeated coats of flatted finish tend to develop a dullness from repeated coats. Ordinarily, I would not expect this after only three coats. It tends to work better to build up to the last coat with gloss and put on the semi-gloss as the last coat.

Brushing poly is a combination of good brush and good technique. Poly is laid on and tipped off. Not so much brushed as put on the surface. Congratulations on using a top-quality brush.

When you sanded between coats, did you sand it flat with a sanding block to give a flat, smooth surface? Each coat's preparation sets up the next coat. A smooth, flat surface tends to help make the next coat smoother.

You might practice brushing: lay it on and tip it off. Just drag the tips of the bristle across the poly to pull it to the wet edge and knock off some of the bubbles that inevitably occur. Poly should go on thin. Thick coats tend to cure poorly because the finish can be trapped beneath a film of cured product. This trapped finish never cures.

The fogginess you mention..... Does it appear that there may be small air bubbles trapped in the finish itself or does it appear to be dull on the surface?

Thanks for your response -- what you say makes absolute sense... the fogginess would probably be better described as hazy. It almost looks dull. I don't think it's trapped air bubbles.

Do you think I need to sand it all off and try again?

Thanks again...