Do you routinely start full dose anticoagulation in new onset atrial fibrillation in the ICU?  

Yes. (Based on CHADS2 + Absence of absolute contraindications, i.e., active uncontrolled bleeding or coagulopathy.)

Case-to-case basis based on the underlying cause and bleeding risk. I don't as readily in patients where it's driven by sepsis: Walkey et al., PMID 27487456