obligation in the past

Yes, it's fine. In your sentence, had to go is in the past tense.

Present tense: You have to go shopping today.
Present perfect: For three days in a row you have had to go shopping.
Past perfect: He explained how, ten years ago, they had had to go shopping.
Future: You will have to go shopping tomorrow.
Future perfect: By this time tomorrow, you will have had to

Isn't it odd to use had to to express a past obligation that was not fulfilled?

Yesterday, you should have gone to shopping, but you didn't.

Saying that there was a need or requirement to do something in the past does not imply or guarantee that that something was done. It only says that there was a need or requirement.

-I really had to win the lottery yesterday.

TeoIsn't it odd to use had to to express a past obligation that was not fulfilled?

In your sentence, Teo, had to has this meaning (from have):
2: have: to feel obligation in regard to

Yesterday, you felt an obligation to go shopping, but you