I found a video explaining that it's fine to hook the new 14-50r without a ground (just connect the 2 hot and 1 neutral) so I did that. I turned it on and everything appears to be working great.
Did this video include the words "challenge" or "hold my beer"? It should have... yes, Youtube is horrible for electrical advice. Come here; we have an upvote/downvote system where a bad answer like that will get downvoted.
Step 1: See if your wires are illegal.
Look at the neutral. If it's insulated white, or a web of bare wires that were wrapped around the hots, that is acceptable. Look at the 2 hots, if they're both NOT white, that is acceptable.
If you have a white hot or a solid bare neutral, that cable was illegal the day it was installed and it needs to go.
If the cable is illegal, the whole cable needs to be replaced anyway. You can't use 10/2 cable (it's obviously illegal) so you must use 10/3 with ground wire. Problem solved.
Step 2: See if you actually do have a ground.
You have to look at the wires coming into a metal box. There may be a ground wire that was terminated at the box metal instead of the recep; that is fine actually. Grounds are supposed to go to the box metal first, and can be carried to the recep via the metal-metal contact between recep and box.
Next, check for metal conduit - if it's 3 loose insulated wires going into a pipe (white or gray neutral, and any other color hots), then you probably have ground via the metal conduit. Again the receptacle can pick up ground via the metal box.
If you don't have a ground wire or conduit...
You have 2 available plays.
#1 you can retrofit just a ground wire. It needs to be #10 and can go to any junction box where a #10 or larger ground wire goes back to the panel, or metal conduit goes back to the panel, or it can go to the bare copper wires that run between the main panel and the grounding rods. Do not attach to plumbing pipe.
#2 you can skip running the ground wire and install a 2-pole 30A GFCI breaker in the panel. Label the receptacle "GFCI Protected" - "No Equipment Ground".
In both these scenarios, you can install a NEMA 14-30 receptacle properly and legally.
Otherwise, it is legal to continue a 10-30 in service
But not all that safe. If there is a problem with the neutral wire, the chassis of the dryer WILL be energized. Now as you know, most of the time when people get an electric handshake, they get lucky and don't die. Do not interpret this as "humans are immune to electricity"; they're not and the wrong set of circumstances can converge to kill you anytime there is an unsafe condition.
The problem with a 3-wire dryer connection is there are no layers to the onion. Mains electrical is designed so there must be a chain of several failures (and at least one code violation) to create an unsafe situation. Not here. You are exactly 1 routine/expectable failure away from a lethal situation.
If you simply can't change the wiring, the 2-pole GFCI breaker + 14-30 recep is the best option.
However if the wiring is illegal, you must change the wiring.
Remember to un-jumper the dryer!!!
The dryer will have a jumper to connect (or not connect) dryer neutral to chassis. You must take care to remove that jumper if you use a NEMA 14-30 connection, so that neutral is isolated from ground at the dryer. (if you use a GFCI breaker, that will enforce this).
I cannot in good faith ever recommend someone connect a machine's chassis to its neutral, but that is what you must do with a 3-wire NEMA 10-30 connection, because the UL approved instructions require that, and that exception was carved into NEC for 3-wire legacy installations of dryers and ranges. If it were me, I'd pursue one of the other options above.