I'd advise against making a sticking board for drawer bottoms. A sticking board is for holding long thin mouldings, not wider boards.
I've found that I like the flexibility of an assortment of scraps and offcuts of various sizes for the sort of job you have in mind. Here are some examples. They turn out to be from box making, as I seem to only have photos of drawers with slips to hold the bottoms.
Here I was grooving a thin strip of pale mahogany-type wood to make a box.
A piece of thinner plywood, which just happened to have a notch cut at one end, is up against a bench stop, held down by a couple of holdfasts. The workpiece rests on the bench, with its near edge just off the edge of it. It's held down and in by the plough plane and its fence. I grooved it in one piece before cutting it into four pieces.
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Here I was making some bigger boxes and had the sides all cut to length first so had to groove them one by one. Again, there's a bench stop and a holdfast and the work is just off the edge of the bench. This time it's wide enough for there to be room between the holdfast and the plough without needing an intermediate bit of thin ply.
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If you don't yet have a bench stop or holdfasts, you could plough your drawer sides by putting two or three strategically placed screws (or nails) into your bench, in equivalent positions, ahead of the work and off to the far side, so it won't slide. Screws can be much easier to avoid than bulky clamps are and can be positioned exactly where they are needed.