Why would my divorce attorney try to direct me to use my spouse’s attorney?

Ask her.. if you do not like her advise hire another attorney.

That is an interesting question. The typical reason that an attorney might suggest both utilizing the same counsel would be to save time and money. This is not a typical situation, however, because the typical lawyer-client relationship is built on privileged information held by both the attorney and the client. When one attorney is representing both sides, this privilege cannot exist. If the facts of the divorce are fairly straightforward and neither side is contesting the divorce this option may make sense.

I have no idea. That sounds very odd. That said, if you are going through a divorce you should strongly consider retaining your own attorney - someone who is going to look out for your interests.