How can I save my bath rugs?

I've heard of a "paintable" latex product that your could apply using a foam brush to 're-new' or make a rug safe. Try looking at hardware stores or craft stores, they may help.

Yes. Here is a link that discusses using a latex product to fix the back of your bathroom rugs: https://homeguides.sfgate.com/repair-rubber-backing-scatter-rugs-101769.html

Most often the backing once crumbly from drying in the dryer are not salvageable unless used in a favorite pets bed. They will certainly love it comes winter. To do anything of inexpensive value, I suppose if you have the use of a sewing machine you might stitch the waffle shelf liners to the bottom of your rugs and buy a little more use from them, then pass them along to your best friend of man ,your pet or your best friend that has a best friend pet. Good luck

Instead of painting a backing have you used those latex underlays? They act as an anti-slip for rag or canvas backed rugs. It looks like a 1" latex grid and comes in various sizes and are folded into a small bag (so you may not have noticed them at the hardware store). Check the rug sections of the store (department or hardware). They should be fairly reasonable since it is a bathroom rug.

Hope that helps!