Last week, my wife was here. The week before that*, I was in Austin. Tomorrow, fieldwork begins. Recently, I've been busy.*I doubt this is correct.
先々週 would be 'week before last'. Otherwise it's basically correct. (Maybe there should be で instead of に when talking about Austin.)
その一週間まえ is not entirely wrong, although I'm not sure it's natural. It sounds like something I would say.I also disagree with Hasa about に vs で. If you are saying you were in a place, に is appropriate.「さいきん、いそがしかったです。」 is slightly odd. It sounds like "Recently, I was busy," not "Recently, I have been busy." Usually you will hear people say 「最近、忙しいです。」 "I'm busy recently," implying a continuing state.The first two sentences with their ここにいました and オースチンにいました don't really sound right to me. There's no grammatical problem, but the nuance of いました is more of "was in this position (or place)." So when you say 妻はここにいました that sounds like she was standing right here, on the blog. I think it would be more usual to say 「先週、妻が泊まりに来ました。」 "Last week [my] wife came to stay [for a while]." or 「先週、妻が訪問に来ました。」 "Last week, [my] wife came for a visit." (Not 100% sure of the grammar of that last one.)Similarly, instead of saying you "were in" Austin, it would be more common to say you "went to" Austin.「その前の週、僕はオースチンに行きました。」 On the other hand, saying 「その前の一週間、オースチンにいました。」 sounds not so strange to my ear. (The difference being "During the previous week" as opposed to "one week before".)