samples of my gelled backgrounds

As David wrote, if you're having issues with spill then replacing the shoot-through umbrellas with gridded softboxes is kind of a no-brainer. But seriously, I wouldn't worry about that since you're getting good results with your black fabric. Just forget about using a white backdrop if you're trying to color it with gels. If you want a less saturated color, you could try a grey fabric, or position the gel so it only covers part of the light.

As for the focusing issue, I think you're overthinking it. It's very difficult for a camera to focus on black fabric. With a subject to focus on, you won't have any issues and the backdrop will be nicely out of focus.

I've taken a number of studio photography classes and in almost all cases we were using f8-f11 to get the right depth of field using either a 50mm or 85mm lens (and its typically the sweet spot of the lens). As has been mentioned above, use a DoF calculator to help you out if you want the background in focus. It would also help if you had a test subject sitting on the stool, a stuffed animal will usually work in a pinch.

An 85mm lens at f8 at a distance of 10ft only gives you a 1'3" dof. At f11 its 1'9"
A 50mm lens, f8, 10ft the dof is 3'10', at f11 its 5'5"
The kicker is that with the 50 you are now further away. To get the same composition, you have to be closer.
So lets say with the 50 you are now at 7 ft at f8, your dof is 1'10" and at f11 its 2'7"

The point, as mentioned before, is that DoF is about aperture, distance and focal length.

Looking at your setup again, I'd swap the modifiers so the one with the black backing that is blocking light from spilling backward from the front umbrella is near the wall. Its easy to see how much light is being reflected back to the wall by the umbrella and then back toward to subject by the wall. At least on camera right there is space between the light and the walls which will diminish the reflected spill affect.

It would be even better to get another umbrella with black backing if that is your intended light source.
I'd prefer a softbox with grid. Larger box and grid will still over wrap when close enough to the subject.

Also, remember Phil's diagram, if you place the source close to the subject then the distance to the background is greater percentage and thus the fall off is greater.

And Curtis is making a good point about staying with dark backgrounds for more saturated colors. White defeats the purpose as pointed out by someone in an earlier post.

If I misunderstood the focus problem I'm sorry, it seemed to me you were wanting to get both subject and background in focus, if that is the case you are running into some optical principles to defeat the plan.
If you are talking only about focusing the camera on the subject, then everything I said about DoF is useless information.

dmward wrote in post #15504429
Generally, when shooting portraits, its intended for the background to be out of focus. I don't remember seeing a post explaining why you've decided you want it in focus.

Because this just looks awful, doesn't it?

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coeng wrote in post #15504756
Because this just looks awful, doesn't it?

As a subject itself rather than as a background, but its so out context without a subject in front of it that I think you are looking at it with a built in bias.

I do have small Nova 16x22 softboxes, should I try using them instead of my umbrellas?

I really am a novice at studio lighting, so please just tell me what you would do. I need ideas. I am experimenting but not getting very far in the process.

gonzogolf wrote in post #15504769
As a subject itself rather than as a background, but its so out context without a subject in front of it that I think you are looking at it with a built in bias.

True, but if I add a person into the shot, then crop it tight, it will be noticeable because there are folds in the fabric and they are out of focus.

coeng wrote in post #15504772
I do have small Nova 16x22 softboxes, should I try using them instead of my umbrellas?

I really am a novice at studio lighting, so please just tell me what you would do. I need ideas. I am experimenting but not getting very far in the process.

This is hard to answer in vacuum. The larger lights will be softer so if thats a priority, then no. If controlling spill is more of a priority then yes.

Regarding not wanting in-focus backgrounds, look how pleasing this painted background is:

https://photography-on-the.net …?p=15102547&pos​tcount=839

coeng wrote in post #15504777
True, but if I add a person into the shot, then crop it tight, it will be noticeable because there are folds in the fabric and they are out of focus.

Sounds like your issue is more with the folds, which is endemic in draped backgrounds. But keep in mind framing tight changes your depth of field so you still need to do tests with a subject.

coeng wrote in post #15504784
Regarding not wanting in-focus backgrounds, look how pleasing this painted background is:

https://photography-on-the.net …?p=15102547&pos​tcount=839

That background is in the same focus as yours, it just isnt an ugly drape.

gonzogolf wrote in post #15504797
That background is in the same focus as yours, it just isnt an ugly drape.

Really, I didn't get that same impression, but you are right about the ugly drape.

coeng wrote in post #15504830
Really, I didn't get that same impression, but you are right about the ugly drape.

That's why you use pleasing backdrops with mottled colors. They have no appreciable detail so they look good at pretty much any level of focus.

What fabric is that backdrop made of?

I would first try stretching it out to get rid of the folds. If there are a lot of creases in it then you might try wadding it up and leaving it in a bag for a few days to create more random creases. The random creases from a wadded backdrop are much less distracting (especially if they're a little out of focus) than a pattern of creases from a folded backdrop.

Every time I buy a new backdrop, I have to rinse it and hang it up to dry, to get rid of the creases.

One can readily see that the posing stool is only about 3' from the backdrop, so it is no wonder that it is not sufficiently out of focus and therefore too distracting!

Furthermore, by being so close, the intensity of the Main and Fill are both insufficiently decreased in intensity at the backdrop by any falloff due to distance!

SUBJECT TOO CLOSE TO THE BACKGROUND...the key sin of most amateurs attempting portraiture. The OP did state "Secondly, I'm limited in my work space ", so circumstances are limiting, in this case.
Somewhat smaller sources positioned closer to the subject would help by increasing the effects of falloff due to distance