Grawss,
There is probably no problem with the game, we may be crossing the communication gap here.
There are two different ways to "abandon" a city.
The quickest way is the new way that appears in the right click menu. If you right click on the city, then the white menu box will appear and in the section with "zoom to city" and "rename city" you also have the choice to "abandon city". If you select this option the city and all its people, shields, stored food, and improvements just evaporates. Before doing this, you should set all the people to be scientists and eat up the food, sell off all possible improvements for cash and then abandon away.
The older and more complex but really effective way to abandon a city is to make sure the city population is at 2 people and then set those people to make sure they are not producing a positive net food output. Look at the city screen to make sure that no surplus food is being produced and this will show if the expected growth rate is either 9999 turns or if there are red shortage warnings in the food bin. You can then disband a small military unit to push enough sheilds into the production bin to complete a settler unit from the build queue, or pay cash to rush the settler, or just wait until enough sheilds get produced to pop the settler out teh old fashioned way. In this case you will get the three choice dialog described by Alphi above.