I have an existing circuit run to the dryer location in an older house that has 10/3 wiring that doesn’t have a ground wire. Just red, black, and white. Can I use that without a ground in a 3 prong dryer outlet.
Yes, a new installation would require an EGC (ground, green), but in existing installations predating that requirement, you are allowed to bond the dryer chassis to the neutral, rather than to the EGC. The proper receptacle is a 10-30, and it's important that the bonding strap connecting the neutral and chassis inside the dryer is installed.
Or perhaps a 4 prong and add a ground wire connected to the water pipes?
A separate EGC can be run, although not to an arbitrary point on the water pipes. Then the proper receptacle is a 14-30, and it's important that the bonding strap in the dryer not be installed. For a list of places to which you can run your separate EGC, see NEC 250.130(C):
250.130(C) Nongrounding Receptacle Replacement or Branch Circuit Extensions. The equipment grounding conductor of a grounding-type receptacle or a branch-circuit extension shall be permitted to be connected to any of the following:
(1) Any accessible point on the grounding electrode system as described in 250.50
(2) Any accessible point on the grounding electrode conductor
(3) The equipment grounding terminal bar within the enclosure where the branch circuit for the receptacle or branch circuit originates
. . .
(the next few options aren't really applicable to your situation)
For (1), on a metal underground water service, only the first 5' of the water pipe in the house is an acceptable connection point. And you should see a ground clamp on that metal water pipe with a wire going to an electrical panel. That wire would be option (2).
However, often it's the case that if you can run a separate EGC, you can just as well run a new 10/3 cable (with EGC) back to the panel, which is a better result.
Cheers, Wayne