Not to be terrible about your question but have you seen about getting an already brailled pill box. Sometimes the tapes are just a pain

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I have had problems with the DYMO tape coming off as well. The tape comes tightly curled in the cartridges which makes it want to curl back up and pop off of surfaces - usually not immediately, but within a year or so.

We have had the best success with this:
https://www.americanthermoform.com/product/braille-labeling-materials-braillables/

It is a clear plastic sheet with an adhesive backing. You can run it through a perkins brailler or an electronic embosser. You then cut the labels apart and stick them wherever you want. These are good for adapting books (because they are clear) and the labels stick great. You really want to use a full sheet at a time if you are using an electric embosser - with a manual brailler / typewriter you can type one label at time since you have better control of the roller.

For the things that need to be in cold environments (like a freezer), or in hot humid places, we have good luck with 3D printing Braille tags that can be rubber-banded or other attached to the object as a label. Obviously, this needs to be for things for which you can reuse the labels (e.g. for freezer tags, "Chicken", "Beef", "Steak" etc.)

I only know of DYMO tape I'm afraid, for the braille label maker. I do too find it a bit awkward to peel but never had problems with it not sticking to things.

Other than that, I wonder whether an adhesive paper exists which can be used with a Perkins or braille printer..?

The only other solutions I can think of include RNIB's 'pen friend' which wouldn't be using braille, or using something like a tacti mark pen, raised letter stickers or numbers.