Finally giving up on St. Augustine. Just won't grow in the back yard which is heavily shaded. I also have dogs who like to scratch when they pee. Any opinions on Zeon vs Palisade Zoysia? Or any other recommendations?

Free pro advice for your area comes from Neil Sperry & one way to get to him is his radio call-in show.
From what you describe, I predict Neil will HIGHLY recommend dwarf mono grass where the dogs are not.
I have no guesses as to what would be best where the dogs are.

The radio show comes on in Aggieland 11am Saturdays.

I've had more luck with Emerald over Palisades in shade. It seems counterintuitive but that's my take.

Never seen or tried zeon.

All depends on the reason it isn't growing right now. I am in the process of switching out my front yard with Zorro Zoysia, solely due to its ability to handle kids constantly playing on it. Both the Palisades and Del Mar that I had used in the past were not able to handle the wear and tear or daily activity. My live oaks are limbed up high enough that light is not an issue so I know the daily activity on it is the issue. You are correct that if it is a shade issue, Zoysia actually needs more light than Augustine (unless you are using Raleigh). But if you think the dog activity is wearing the Augustine down faster than it can grow back, I would say definitely give Zoysia a try. FYI I am getting my Zoysia from King Ranch if you are looking for a source.

His second sentence contained "yard is heavily shaded". Thus my response.
Grass can generally overcome one obstacle, but not two or more. Shade is one thing. Shade + traffic - not aware of any grass grown in Texas that can handle the two together.
Generally, you can have grass or trees, but very hard to have both do well.

I grow grass for a living and have 5 different types of grass in my yard, yet I still have an area that I have had no luck with.
I have Palisades in my north facing front yard under big trees. It does fine except really close to the trees. In fact, it has overgrown common bermuda outside the drip lines.
I have a tiny bit of 419 between the sidewalk and the street. It has been there for close to 20 years and does just fine.
All the common that was originally planted in the yard(23 years ago) is almost gone. It has held on in full-sun areas.
I have Diamond zoysia in the back yard and it does great until it hits heavy shade,
I tried Palisades in the shadier areas of the back yard and it has survived where it gets some sun.
I finally gave up and planted some St. Augustine in the worst shaded area - a 24" burr oak on one side(east) and the neighbors 10' fence and my 3 18' crepe myrtles on the other. It is going to get a little more sod this year, but if it doesn't make it I am going to seed fescue in the fall (and probably every fall after) - I don't have a traffic issue there.
I tried Celebration bermuda - everyone claims it is a more shade tolerant bermuda. Problem is, it is still bermuda and has to have 6-8 hours of sun. I planted more Diamond where the Celebration was and it has done well.

I really like my trees, but they have caused havoc with my turf.

Wait, I'm getting ready to replace my tif419 in my backyard with zoysia in the next month. A quarter of my backyard probably gets 4 hours of sun a day or maybe a little less, and that area is basically dirt at this point because the Bermuda can't handle that level of shade.

I've got a place near me that sells Palisades and Zeon, both same price per pallet. I was assuming the Palisades would be better shade tolerant but that's not the case??

I've also got a female dog and couple teenagers and so the yard gets a little bit of traffic but nothing extreme. I'm in Fort Worth so my "soil" is pretty much minimal in some cases, and I've got some limestone rock in certain areas that isn't too far under the surface. I guess the soil is considered part clay over here in Fort Worth?

Which of those 2 varieties of Zoysia should I go with? I also don't have a reel mower, so I'll be using a standard rotary. I was under the impression the Zeon would need to be mowed low with a reel for it to do well.

In general, the finer bladed zoysia actually does a little better in shade than the wider blades. I have never planted any Zeon so I can't give you any first-hand info on it.
Four hours of sun will not be sufficient for zoysia. It will do ok year one. By the end of year 2 or at least part way into year three you will probably have to resod the shadiest areas.
If trees are your cause of shade, find a reputable arborist and get them trimmed as high as they will go and thinned as much as they can. Do not let them prune the interior growth leaving "lollipops" on the end of branches. Make sure they are actually thinning the canopy.
If your house is causing the shade, start thinking of alternate landscape plans for those areas. Or, you might get a little more time out of St. Augustine. It will "get by" with 5-6 hours of sun - maybe 4 hours direct and another 1-2 of mottled sun. Morning sun is more important than afternoon sun.