I quit using premade tack cloth years ago---same reason as you have found.

Now, I do this---vacuum/blow off dust, using a blue paper towel(the HI/hardware store kind) I dampen the towel with the thinner used for the finish I am going to use on the project and wipe it down thoroughly.

Example---varnish---use paint thinner
water based poly---use water
Shellac---use water and let dry completely(alcohol is too expensive to use)

hi, I have used a oil based stain, so I don't think I want to wipe it with thinner as it will remove the stain.

I need something slightly abrasive but not so much like sand paper, like a screen to remove some of the the nubs, and dust and bubbles that occured during staining.

I am thinking of using a plastic 20% shade cloth, it's a net that roughtly look like the tack cloth without the tack. I don't want to remove the pigment on the surface and I think it will rub out the little particles, or if it does abrade, at least it is very slow and controllable. Do you think that will work?

In my experience, a light sanding(400 grit or 000 steel wool/equivalent plastic stuff)removes not only the 'nubbies', but also the excess stain---which winds up in the brush/finish being used.

Now, I seldom sand before staining with more than 150 grit paper---as any finer sanding burnishes the wood and prevents stain from being absorbed as deeply.

Sounds like a lousy tack cloth.

They need to be sticky, but not enough to leave anything behind.

There can also be compatibility problems between tack cloth treatment and some finishes.

A clean lint free rag moistened with the appropriate thinner does a good job without compatibility issues.

Using water is almost always a problem though.
It will raise the grain requiring a light sanding, leaving more dust.
If you repeat the grain raising - light sanding cycle a few times you get less raising each time, but using denatured alcohol to wipe down before water based finishes does the job.

While the alcohol is more expensive than paint thinner you do not really need that much. The rag only needs to be barely damp.

Paint thinner can leave enough petroleum distillate behind to cause problems with water based finishes (paint thinner is a rather 'dirty' chemical mix).