Well,..in my zone 3, the Latham is the first producer, small berries but good flavor.
Not sure what you might do,...I would leave the canes alone after picking and remove the following spring,..it might put too much energy into the remaining canes for them to push flower.
You might need to neglect them a little?

One question I have are the primocanes fruiting like fall bearers? i.e. only from the upper third? What's going on makes little sense as even fall bearing cultivars can give a decent summer crop. My only thought is they are so stressed, they fruit. Try feeding them more, use compost too every spring. Mulch with compost.
If you start over try another cultivar. Taylor, or Encore, or both! I only have a few summer bearing. Cascade Gold, Taylor, Encore, Prelude(which does give 2 crops, why it is considered summer unsure?). Oh I also have Honey Queen. I mention Taylor and Encore because they give nice crops and withstood last winter well. Encore seemed not bothered at all, but does give a very late summer crop. The others all showed some degree of damage. Maybe try three or 4 different summer bearing and see what works best.
Another thing that could stress them is a virus infection. If they did well at first, and then starting doing this, maybe they caught something? I would not plant in the same spot.
Any leaf stress? Veining of the leaves? Yellow leaves? Discolored leaves? Do they get full sun? They need a fair amount. Can produce with 5-6 hours prefer more.

Thanks Konrad and Drew. Went out and took a close look and it appears that at least 75% of the canes have berries now. The other 25% are small canes, maybe 3-4' tall. The fruiting canes are anywhere from 7-9' long. All the berries are on the very tips of the canes. The patch is in the sun, no shade at all. I fertilize in early spring and again when they start blooming. I keep them mulched with a combination of straw and mulched leaves over the winter and then add more mulched leaves as needed to keep about 4" of mulch on them.. I did see 2 or 3 small (2-3') dead canes and a few dead leaves low on the longer canes. Some of the berries are turning red now so guess I'll pick what I can before frost, leave the fruiting canes and if they leaf out in the spring I'll prune them back to about 5' and hope they produce laterals for fruiting. That didn't work last year but last winter was pretty rough so that might have killed them. The winter killed all my hydrangeas down to the ground and about 1/3 of my blackberries. Right now I'm wishing I had chosen a different variety but I've never known Latham's to do this.

It is very strange, and you seem to be doing everything right. Some dead leaves down the cane is normal. They are fruiting like everbearing or fall bearing. Since you got them from different places, seems strange. Unless those nurseries all got them from the same wholesale nursery?
Seems very doubtful. Maybe they are not Latham? Some fall bearing types have poor summer crops, or have poor crops in some areas, decent in others. When it walks like a duck, talks like a duck, they certainly are behaving like fall bearing plants.
Mail order new plants from Nourse farms
http://noursefarms.com/
One of the best possible places to order anything, these are top rate, it doesn't get better!
Or Indiana Berry
http://noursefarms.com/
As good as Nourse. Very cheap, huge canes. Most cultivars are sold in the spring,

My only other suggestion as you didn't mention it is to fertilize them too. if money is an issue, miracle grow works. Not that expensive, goes a long way. Or go find a feed store and use cottonseed meal or alfalfa meal. Cottonseed is probably better for raspberries. Gives a little acidic edge. Raspberries like slightly acidic conditions.Holly-Tone is good for them too, more expensive.
I had an exceptional day today with my raspberries. The fall bearing are really producing like mad. I usually get 60 berries a day, but today got over 100. a good day! Mine produce in time and I'm in 6a/5b Looks like you are in 6 in KY? They should finish before winter there, that is strange too!
I grow berries of every color
{{gwi:48150}}

I saved these from the summer crop
The black raspberries are in the jam jars at this point. I still have the rest frozen for this winter. I love coating them in chocolate! The best!
{{gwi:44117}}

I'm in zone 4 in Wisconsin and my Latham are acting the same way. I planted 2 bushes last spring and picked some berries this summer. The bushes grew very well this year and small new plants have cropped up. The two original plants are loaded on the ends of all the new branches with berries that I assume will not have time to ripen...but hoping just the same. I guess I'll prune the summer bearing branches to the ground and leave all the others until next spring. As a new raspberry planter, I'm confused, but hopeful.

Well with fall bearing types that fruit on the top 1/3, you can only remove the top 1/3 and they will bear in the summer on the lower 2/3's. Maybe you should try this?