You should try to get past this crisis with only clearing a blockage, although eventually the cast iron will have to be replaced. If your house was one of a group of houses of the same construction, you could locate your cleanout by looking at other houses of the same plan in your neighborhood. I have helped several people in my neighborhood find their cleanouts by doing this, by finding one that is not covered with soil. Then probing with a long thin screwdriver for the buried one in the corresponding location. Neighbors will know whether the city sewer is in the street or elsewhere.
My experience in Dallas and elsewhere is that the city sewer is under the street in front of the house. In some places there is a mark on the curb (often a "Y") indicating where the house drain sewer connects to the city sewer, but this will not in general line up with where the drain leaves the foundation. The drain line will come out 3 to 6 ft and then angle off at 135 deg (or 45 deg depending on how you look at it) to line up with the city connection and then have another 135 deg angle to go to the connection with the city sewer.
In my house, on a slab built 1970, the main clean out is in front 3 ft from the foundation. This is a double cleanout (two adjacent 3 or 4 inch screw fittings in line) so a snake put in the near one goes out to the city connection, a snake put in the farther one goes under the house. A newer type of cleanout is single and the operator of the snake must perform manipulations to direct the snake. Our house has another cleanout on the back wall of the house near the kitchen sink. The screw caps in these fittings (brass or plastic) can be stuck and may have to be destroyed to be removed.
Presumably the sewage emerging from your shower drain is from your own house and the blockage is in your own drains, and your responsibility. However in Dallas the city seems willing to fix sewer collapses involving the Y connection and even close to it. You might check with city or county water and sewer. But if there is a blockage in the city sewer and you are the last house before the blockage then sewage from other houses will back up into your house. This would not be your responsibility. Are you the lowest house on a hill or the last before a city connection?
Two houses in my neighborhood that are each the last house on their street at the bottom of a hill had sewage emerge from shower drains and cover the floors (episodes 30 years apart). In each case the cause was a blockage in the city sewer where the city sewer under the streets connected to a larger sewer line running perpendicular across the base of the hill. The city installed large cleanouts at the end of each of those two streets, and this provided a relief overflow when another blockage occurred. Sewage flowed out into an open space in a park rather than into a house.