The last time we . . . . . . . . dinner with them, they . . . . . . . . for a new house.

In the sentence, "The last time we had dinner with them, they were looking for a new house," the past simple tense and past continuous tense are used to describe two actions that occurred in the past.

The correct form for the sentence is "The last time we had dinner with them, they were looking for a new house."

The other options are not suitable because they do not use the correct verb forms or tenses:

Option A: have had/looked
This option uses the present perfect tense and the base form of the verb "look," which do not match the past tense context of the sentence.

Option B: were having/have been looking
This option uses the past continuous tense with the present perfect continuous tense, which is not consistent with the structure of the sentence.

Option C: have been having/look
This option uses the present perfect continuous tense and the base form of the verb "look," which are not appropriate for describing past actions.

Option E: have/are looking
This option uses the present perfect tense and the present continuous tense, which are not suitable for describing past actions.

Therefore, the correct answer is Option D: had/were looking.