I hate the notion of treating soil with fungicides if a fungus problem hasn't been verified. They certainly have their place in a gardener's arsenal of tools. I just always want to start with the least aggressive treatment available that proves effective...so start small and go big if necessary. Not preachy - everybody gets to choose how they garden!
I am making huge assumptions here because we haven't a full picture of the problem (either figuratively or literally) as the photos are too cropped - but it appears you are growing multiple peppers in something like a window box? Not much soil for roots to develop...and if they were filled with store-bought soil also less likely to have a soil-borne pathogen in year one.
They also appear possibly to be something like dark plastic? In Houston, in the sun, you could easily be frying the plants roots...if you suspect this might be the case insulate the 'pot' by wrapping it with something as simple as a few layers of canvas sacking.
Soil looks to be super saturated - but it also looks like you had just watered as there are droplets on the pot edge or support - are you checking to see if the plants need water before applying more? And of course watering deeply but less often is great...but only if you also have adequate drainage so I am assuming there is drainage?
Also, helpful if we knew more about their orientation; sun, etc...more info means better suggestions.
Consider taking pictures and a freshly removed leaf or two to your county extension office for help - they do great work (check online for schedule of when they offer ID and pest and disease assistance).
Just food for thought - good luck.
