Am I able to wire a 3-Prong dryer outlet with 8/3 Romex?

You could. But don't!

Instead, replace the 3-wire outlet with a 4-wire outlet. It is much safer. There are no downsides. You will have to replace the dryer cord/plug with a 4-wire cord/plug, and remove the jumper connecting ground and neutral on the dryer (typically a green wire going from the neutral connection to some bare metal spot on the dryer).

You would have to do the same thing if it was a 10/3 as well. That is, assuming your 10/3 is (like the 8/3) 3 regular wires (typically black/red/white) + a ground wire.

If your 10/3 does not have a ground wire, ask for more help.

3-prong outlets aren't "your dryer outlet". 3-prong outlets are an obsolete and dangerous form of outlet that's been outlawed since 1990, but has been grandfathered in old installations. Because of this grandfathering, and the fact that the home stores sell them as unit replacements for broken sockets, people just keep installing them.

Stop.

You must convert this setup to a 4-prong outlet. It's not optional, it's a Code requirement.

You must also change the dryer cord to a 4-prong cord. Google the dryer instructions and they will tell you to remove a bonding strap connecting neutral to the dryer's chassis/ground, and then wire up the 4-wire ground wire to chassis, and the other 3 wires just as before. (with the bonding strap removed!)