As someone who has just sourced #14 wood screws (probably still too small for the original hinges) for our front door, I hate dealing with old, slotted wood-screws for similar reasons.
I often use heat first - a kitchen blowlamp is good for this (creme broulee anyone?). I like MAPP gas, but it's hard to get a small enough flame with the burners I have. If you want to save paint, or simply not inhale it(!), mask off around the screw plate with a bit of tin or similar.
I've read many times that you should use a soldering iron on the screw head, but this is nonsense. It doesn't get hot enough, solder doesn't wet rusty steel, so the conduction is poor (I can't think of a better heat transfer method incidentally), and, as you have to heat the rusty screw, some of the hinge plate and a cylinder of wood surrounding the thread, you'll struggle to find an iron with the necessary wattage.
Anyway, cook the screw head as much as you can, then let it cool thoroughly. As above, use a sharp screwdriver that fits exactly, and clean out the slot before even attempting anything. It should move cleanly, as you'll have reduced the rusty wood surrounding the screw to crispy charcoal. At that point drill out and plug.
The above was a non-starter with our #14s anyway, as the leaves of the hinges are around 3/16" thick cast steel - either the unscrewed properly or we would have to drill them (and risk them being permanently stuck!).
I note in passing that there is an issue with dowels: they're long-grain. So a traditional woodscrew will struggle a bit, and will never grip as strongly as into the original wood. Making a cross-grain dowel somehow probably won't help much, so...
... we tried a slightly different tack this time, and remember this is for traditional, and pretty big, woodscrews, with a chunky thread, not modern thin-shank hardened screws, so YMMV.
We used softwood for the dowel, on the basis the thread would cut a bit better, and we pilot-drilled and split the end of the dowel (like a Rawlplug). Titebond 3 was used to secure the dowel before any attempt to drive the screw, with the split across the stile's grain, (not along it). The pilot hole went all the way, with a wider bore for the unthreaded part of the screw, just enough so it would easily still cut a thread to pull itself in.
This seemed to work well, and we have the second hinge to do later today, now the weather is nice.