CalifJimso most likely they did not go shopping because they didn't have to.

Yes, you are right. I stand corrected. 'The speaker would have had to go shopping.' will normally mean the speaker didn't have to go shopping. And to label the first one as 'before shopping' is also not normally or most likely correct. Most probably, the speaker thankfully didn't have to go shopping.

CalifJim 'have had' is the present perfect tense, and you can't use that tense with a mention of time uless it's current time.

Wow, this is new to me! Thanks very much. Otherwise I would have wrongly said, 'I've had to go shopping this morning.' in the afternoon or evening! The 'present perfect' requires the (mention of) time to be present. So, to test if I have really understood:

I am tired this afternoon because I had to go shopping this morning.

I am tired today because I have had to go shopping this morning.

It has been a pleasant morning so far. I have had gone shopping early in the morning, too.