I would say yes it is synonyms however remember to class into informal, semi formal and formal… I would never use obligations referring to ironing- besides I ditched ironing when my kids arrived 🤭

This is more formal than what I would say. Formal: I have some chores to take care of. Informal: I have some things to do at home. VERY informal: I have some sh*t to do.

Yes, you can say 'obligations at home,' this would be very general but would include household chores, housework, taking care of children/elderly, cooking, cleaning, doing yard work, or anything else like that.