The Appendix is very explicit (note use of the word "may"):
Urgent wish for children: You may either build a Dwelling on an empty Cavern by paying its building costs or take 3 Gold from the general supply. If you do the former, you may then carry out a Family growth action. (You may not grow your family on this Action space unless you use the first action to build a Dwelling.)
Earlier, the Appendix says the following about "and then":
You must carry out the first action. Only afterwards you may (but do not need to) carry out the second action. You may not carry out the second action without carrying out the first one first. We use “and then” instead of a simple “and” to indicate that the actions are not carried out at the same time but one after another.