You can have any number of 30A receptacles on a 30A circuit.
As long as they don't serve certain loads for which dedicated circuits are required (EVSE's come to mind). "Dryer" is not on that list.
Unfortunately, you own one of the two residential dryers in North America that take larger than a 30A circuit. I doubt that; Occam's razor suggests it's really manufactured for 30A breaker/10 AWG Cu wiring, however, you were preceded by a "nitwit" who liked to oversize breakers.
I suspect that consulting the dryer's nameplate, and its installation instructions (find model number; use Web) will affirm it's made for a 30A circuit.
If that's all true, please - correct the breaker, particularly if the wiring is only #10.
First, correct the dryer circuit to a full 4 wires.
Unfortunately a lot of dryer circuits were installed 3-wire, with a 3-prong plug. "what's wrong with that?" Dryers use 240V but they also use 120V - so they need 2 hot wires and a neutral. But you want ground, and you need it for a circuit extension.
From 1966 to 1996 they allowed 3-wire dryers and ranges, mainly to soak up existing stocks of wire that was otherwise illegal. That included /3 NM (black, white, red, no-ground) and SE cable (black, black, mesh neutral). Once those stocks exhausted they were to use /3 w/ground. Unfortunately some used /2 w/ground (abusing ground as neutral, which it isn't insulated for); that was never legal and must be removed on sight.
If the installation was legal, you can retrofit an actual ground.
Converting the dryer to 4-wire is trivial: google the instructions and they will show what to do.
Here's where I'm fuzzy on Code: Can you extend a circuit with a retrofit ground? I argue "yes". If not, change the dryer circuit to 10/3 w/ground.
Then you can extend the dryer circuit to another 30A receptacle. It doesn't need to use neutral, so a NEMA 6-30 socket can be used. A NEMA 10-30 can never be used; all NEMA 10 sockets should be smashed to prevent re-use.
Down the rabbit hole of NEC Article 430 - motors.
The saw is a motor load, but there isn't a separate article carved out for it (as there is for air conditioners, 440). This section gets pretty weird.
You didn't give us the nameplate data, so it's impossible to do a proper analysis. But most likely:
- You need to derate by 125%, so "13A" becomes 16.25A and you must put it on a 20A circuit not a 15A circuit.
- Notwithstanding nameplate data, You are probably allowed to go to a larger breaker to prevent nuisance trips, within certain limits. 30A is most likely within that limit.
So if you were hardwiring it dedicated, fair chance you could wire this with #12 on a 30A breaker. Really.
However, since you'll be plugging it in, that doesn't really work. You could plug in anything which means 240.4(D) kicks in, requiring #10 wire on 30A breakers. Which requires 30A sockets. 210.21. And multiple sockets allowed on 30A. 240.something.
So yeah. Extending the dryer circuit is probably legit.
Has the red queen lost her head?