TLDR: You can easily and safely get the charge speed you want, charging your Tesla at 150 miles per 10 hours - 3 times the speed of the 120V charging that you tried. You can do it with 12/3 wire, a 20A 240V breaker and the socket at bottom. You just need the correct circuit and dongle.
30A is overkill and I wouldn't waste money chasing 30A. But feel free if you want to; just do it right.
There is no limit to the number of receptacles on a 15-30A circuit
You don't actually need a 30A circuit, but you absolutely can have a 30A circuit that has both a TT30 outlet and a NEMA 14-30 outlet! You simply need to use a 2-pole (240V factory handle-tied) breaker.
"That was easy"
Wait. One of those is 120V and the other is 240V. Can they mix?
YES! 120V outlets and 240V outlets can be mixed on a circuit, as long as the amps match. (30A on 30A; 20A* on 20A etc.) Doing so makes it a "MWBC" (not really important what that is) - the important part is, it needs a factory-made 240V breaker with a feature called "common trip". As long as you have the correct breaker, you're all set.
See the bottom for a factory made, UL listed socket that mixes voltages.
They make a correct socket for ANY application! Really.
You're already familiar with the TT30. Meet the rest.
Above are 120V and 240V sockets for 15A to 20A.
Above are 240V sockets at 30A. Do not use the one in the middle. If you're wondering where the 120V/30A socket is, you already met it. It's the TT30. Despite its slant eyes, it is safe.
Why does it matter? If you use the correct socket for the voltage and breaker, then you won't kill anyone or set anything on fire (or have annoying breaker trips).
The car comes with a "travel unit" EVSE that is intended to live in your trunk and be used for opportunity charging "on the road", which is why they give you the plugs you'll find on the road e.g. the large RV socket (that's An RV socket not an EV socket!) However, Tesla makes 8 of them and will sell you any for around $40.
Note they don't make a NEMA 6-30 - use the NEMA 14-30 instead.
However the NEMA 6-20 is perfectly adequate for your needs. It will provide 3 times the charging speed of the 120v charging that you tried and weren't happy with.
Circuit breaker (and wire) MUST match the socket
On all circuits, the breaker must be an exact match for the socket size. So 30A breaker = 30A socket. Don't ever put a TT30 on a 20A circuit ever again. If you want to install a 120V/20A circuit, use the NEMA 5-20 which is made for that, and then make a short "dongle cord" to adapt from 5-20 to TT30. Keep the dongle in your trailer so you can plug in anywhere. (actually you can just buy that at RV stores).
- The only exceptions are that 15A sockets are allowed on 20A circuits - that's because all 15A appliances and sockets are certified to not fail dangerously on 20A circuits. Also an exception for 40A circuit but nevermind that.
Notice something special about it? That's right. The sockets are different. And now that you have the decoder ring, you can see one is 120V and the other is 240V.
You feed that with 12/3 cable off a 240V "2-pole" breaker with "common trip". Now you can charge your RV and your Tesla at the same receptacle - in fact they can be plugged in at the same time since they use different sockets. (just don't use them At the same time).
Now do you see why it's important that the right socket be used for the right job? When you use sockets properly, you never need to worry about plugging the wrong thing in and killing someone / starting a fire.
If you must do something improper, do it in a short "cheater cable" that is 6" long - that way people who pick it up look at it and go "well hey, this is a cheater cable! I better be careful how I use this!"