Old English sceal, Northumbrian scule "I owe/he owes, will have to, ought to, must" (infinitive sculan, past tense sceolde), from *skulanan, a common Germanic preterite-present verb (along with can, may, will), from Proto-Germanic *skul- (source also of Old Saxon sculan, Old Frisian skil, Old Norse and Swedish skola, Middle Dutch sullen, Old High German solan, German sollen, Gothic skulan "to owe, be under obligation"). This is said to be related, via a past tense form, to Old English scyld "guilt," German Schuld "guilt, debt;" also Old Norse Skuld, name of one of the Norns.
These Germanic words are reconstructed (Watkins, Pokorny) to be from a PIE root *skel-(2) "to be under an obligation." The basic sense of the Germanic word probably was "I owe," hence "I ought." Cognates outside Germanic include Lithuanian skelėti "to be guilty," skilti "to get into debt;" Old Prussian skallisnan "duty," skellants "guilty." But Boutkan gives the group no PIE etymology and writes that the alleged root, limited as it is to Germanic and Balto-Slavic, "is likely to represent an innovation on the basis of North European substrate material."
Shall survives as an auxiliary. The original senses are obsolete; the meaning shifted in Middle English from obligation to include futurity. It has no participles, no imperative, and no infinitive. Its past-tense form has become should (q.v.) and has acquired special senses of its own.