Is there a way to make or get Band-Aid tape? I want a skin-applicable adhesive that works without the absorbent patch on it. Is there another similar tape or a way to get this kind of sticky adhesive without cutting up a dozen band-aids?
Leucoplast - comes in rolls in two different widths. Sticks well, no padding anywhere, just a roll of tape, and can be cut to whatever size and shape you want. Alternatively, ordinary zinc oxide plaster tape comes similarly on a roll, no padding, but often doesn't stick so well, particularly if you're cutting very small pieces to use.
I personally hate bandaid material, but here are possibly some alternatives:
Scotch Tape. This is a mild tape, that has good adhesive qualities when not in greasy, wet or similar environments. This should not make your skin irritated.
Painters or similar tape also works. This tape has better adhesive than scotch and bandaid materials. Also, faces better in greasy, and wet environment.
There are several types of porous medical and surgical tapes on the market. Band-Aids are made of different materials like latex or fabric, so I'm not sure which kind you are looking for. Most of the major brands sell something. A good medical supply store or drug store will have some choices.
Here are a couple of manufacturers. You can explore their offerings to get a better idea of what you want.