This question was raised by a friend of mine on discussion, and I thought i'd briefly answer it in the form of a post. A common misperception upon reading Being and Time is that somehow being-in-the-world is more originary than intentionality. Heidegger in fact grants this ontological privilige to being-in-the-world during his extensive discussion of zuhanden and vorhanden modes of comportment, but the crucial point is missed if we merely take 'fundamental' to mean more originary, specially if we carry a traditional notion of intentionality (which Heidegger rejects in his lecture courses 1925-1927, specially in the Basic Problems of Phenomenology).
I'm not sure whether 'more fundamental' would be the merrier expression. Husserl had an altogether different idea in mind when he spoke of intentionality: he thought he could map the entire constitution of thelifeworld in terms of the consciousness of the subject. According to him,through the phenomenological reductions, one could gain access to 'thingsthemselves', the phenomenal correlates of the mind as they are given. What was left for him to discuss, then, was a series of formal and regionalontologies in which he could map out the different kinds of objects thatcould appear before the mind, as well as their correspondent 'modes of presentation'.
Heidegger rejects the thesis that we can explain the constitution of thelifeworld by starting with the subject, only to then proceed to explainthe world. He thinks that the real question is not what objects arepresented to the mind, but how do objects and ontologies first becomepossible. According to him, it is phenomenological suicide to transfer thebulk of intentional life to a conscious subject since what is primarilygiven are not objects as correlates of some act (Vorhandenheit), butengaged agency in everyday practices (Zuhandenheit).According to him, both 'modes' of being are different ways of dealing withentities, and thus constitute different kinds of intentional comportments(Verhalten).
In this view, one is not first and foremost a mindconstructing a world out of intuited objects, but rather one is firstimbedded in a world of holistically familiar meanings in which one doesnot posit entities as objects in theory, but puts them to use. The entireidea is that our ontologies, the way in which entities get determined asobjects, will depend entirely on the kind of uses and practices into whichwe are imbedded. This ammounts to saying that there ISN'T a possibleontology that just gives a description of possible objects of experience.To say this is to give up on the Platonic ideal that there must be a sortof theory underlying all possible forms of discourse, and which can serveas foundational for all sciences.
Heidegger thus reshapes the traditional concept of phenomenology and says that what we must start with is not aworldless ego or mind, but everyday practices into which we are embedded.In a way this answers your last question, since Heidegger is implying itis impossible to have pressupositionless phenomenology; one can point outat the different kinds of relations Dasein has to entities and how theseoccur as ways of existing, but one cannot have a theory of all theories.One cannot explain engaged agency by appeal to some subject-objectrelation since one cannot spell out a set of rules which are unconsciouslyregulating behavior, and which phenomenology can make explicit. To expectthis is to think one must pressupose tacit theoretical guideliness inorder to build up a meaningful world or practice.
Heidegger thinks it's the other way around: we are first and foremost in aworld in which we share common practices and ways of speaking, and theoryis only derivative. He wants to avoid trying to ground all kinds of beingin a causally self-sufficient source, as Dreyfus points out. There are noregional ontologies to give since there is no single way in which objectsmay be given or understood. Thus, Heidegger does not 'drop' intentionalityas much as denies it must be understood in terms of consciousness. To quote Heidegger:"The everyday way in which things have been interpreted is one into whichDasein has grown in the first instance, with never a possibility ofextrication. In it, out of it, and against it, all genuine understanding,interpreting and communicating, all re-discovering and appropiating anew,are performed. In no case is a Dasein untouched and unseduced by this wayin which things have been interpreted." (213) (169)
What is being asked about is then about this pretheoretical,non-subjectivist understanding of being, of relating to beings and theways in which these modes of being are given. This will, as it turns out,imply we cannot simply interpret objects as the correlates of some privatemind:"One of our first tasks will be to prove that if we posit an 'I' orsubject as that which is primarily given, we shall miss completely thephenomenal contentof Dasein." (72) (46) This should do for now.