What’s a good way of detecting drafts

As 'cheap' is one of the requirements -- I'd probably go with a random orbit or even just a vibrating palm sander. You can get some decent models for $50-60.

I would recommending one of the 1/4 sheet models, where you can use any sand paper that you cut down to size, rather than the fancy shapes that require you to specifically buy sand paper made for that model. (the circular random orbit might be an exception ... the holes help to clear out sawdust faster and keep the paper from getting bogged down as quickly, but if you're going for cheap, 1/4 sheet's still your best bet in the long run).

I'd loosely secure the whole bundle with a ratchet strap or something similar, jog the slats so they're well aligned (stand it on end and tap it on the ground), then tighten the strap, and sand the edges down.

If the slats are really thin, and you fear them breaking, either add another strap close to the edge, or even just bundle it up with tape. (if you don't want to deal with tape residue, use cling film or similar first, then tape it)

You don't say what saw you used to cut the wood, but if you use a tenon saw you should get a cleaner cut in the first place that requires less sanding. This is because the saw has finer teeth for a smoother cut and a stiffening spine that means you get a straighter cut.

Clamping a second piece of wood to use as a guide is another alternative.

Alternatively you could use a plane to remove the excess wood - but make sure that you clamp a piece of scrap wood to the end of the plank to avoid it splitting:

------> direction of plane
+---------+---+
|         |   |
|    ^    |   | <- Scrap
|    |    |   |
|    |    |   |
  direction of grain