Ah! Good point! This problem is super interesting then! You've raised an interesting and neat point about this that I had not considered the implications of them seated linearly. In that case I think you are on the right track with reasoning. Consider the family as a unit which can sit in any of 10 possible positions. Now we will examine disjoint cases for each position. We will consider the two couples simultaneously because we can just multiply the solution by 2 because we can interchange the couples' positions (i.e. couple 1 can sit in chair pair A while couple 2 can sit in chair pair B and vice-versa). Consider the first scenario you have laid out, FFFXXXXXXXXX. We must then consider the scenario where the first couple is all the way to the left, FFFc1c1XXXXXXX. Couple 2 has 6 possible positions. Notice that moving couple 1 one slot to the right allows couple 2 5 positions only. Likewise, an additional spot over allows couple 2 6 positions again. In other words, any time there is only one space left between the family and the 1st couple, the second couple may not sit there. This occurs for 2 of the possible positions of c1. From this line of thought we realize what we have to be concerned about is placing the family and the two couples and in what scenarios we have single seats. Notice now that the number of positions for couple 2 is determined by the total of all (length of a string of X - 1). That is why leaving single seats reduces the total as such. How many ways can we arrange a 9 character string consisting of characters FCXXXXXXX such that there is a length of 7 X? 4. How many can we such that there is a length of 6 X? 8. The math will continue. From this logic and our equation above we can deduce how many total arrangements there are for the family and the couples (remembering thus that you must multiply by 2 for symmetry between the couples). Then that answer is multiplied by 5! for the remaining guests, and 16 for the intracouple and intrafamily permutations (224). Does that make any sense?

As a side note, this seems like a question that might be too rigorous for an exam if the exam is targeted at new learners of this material or if the exam has several questions/time constraints. Would this be better suited as a homework problem?