If the global temperature dropped could the southern coastal cities...?

If the jet stream moved south by 300 miles. And the cold water rose above the alution breach under the ocean. Forceing cold water south. Then Southern Cal. could have snow and the rain from such. But that need come fron the north not south. What you are asking is what would be the weather efefect on AZ. And NM. there it would change the weather paterns.East side of the Rockys. More than CA.

If the global temperature dropped, the southern coastal cities wound not get some type of lake effect snow (LES) with the Gulf of Mexico as the provider for moisture.

LES happens when arctic air blows across a body of open water.

If the Gulf of Mexico were the source of LES moisture, the arctic air flow would have to come from the south. There's little if any chance arctic air will ever come from the tropics.

Well, if the global temperature sinks, anything happening in the north could happen in the south. I don't understand your point with the Gulf of Mexico because if it is rather "warm" water now, it is because it is part of the north Atlantic gyre, turning around the Azores, thus bringing tropical water in the bay. If the global temperature sinks, so will that of the gulf, don't you think?

"Lake effect snow" and "lake effect fog" are called, advection snow or advection fog, elsewhere in the world. the American "lake effect" is used because of the Great Lakes. Air displacements are called, convection (up) subsidence (down) and advection (horizontal).

When the geostrophic wind blows onshore along a shoreline, it brings air from above the water, over the land. Especially during the end of the fall, the water can be much warmer than the land. As that moisture rich air moves over much colder land, it causes an adiabatic cooling resulting in condensation into fog and sometimes, drizzle. If the temperature is cold enough, light snow may appear and that is what is called, "lake effect snow" on the shore of the Great Lakes of the USA and Canada.

But the same appears e.g. along the Oslo fjord, in Norway, where I live. Especially fog. And that is why the Oslo international airport of Gardermoen is far north of the city, instead of on the shore of the fjords as it was before. Too much advection fog to keep the airport open at all time!