The philosophers I study under criticise the sciences for not being rigorous enough. The problem goes both ways. The sciences often do not understand the basic concepts from which they are functioning. A good scientist will also have a rudimentary understanding of philosophy, in order to fiddle with the background epistemology of their work.

You are correct in thinking that Continental philosophy is not continuous with the sciences, because it is the core of the humanities and as such being continuous with the sciences would be unnatural for it. I still think that asking questions about our connection to existence is interesting and important, although I personally do not find Continental philosophy as potentially fruitful as Analytic.

Intuitions are by no means accepted within the discipline as a whole, and are also an interesting topic of debate within it. Because philosophy is a highly speculative discipline it isn't going to be following a normal scientific model, but instead will model constant discovery. If you want to see where science connects up with philosophy what you should look at is the disciplines that end up coming out of philosophy as questions that can be answered scientifically. This is what we produce with regard to science.

Philosophy is the core of the academic disciplines. It isn't in the business of scientific inquiry and it should not be. Some philosophers are still looking for universal truths after all. Simply disagreeing with the idea of a priori does not make it go away.

It is good that you recognise there are problems in philosophy. Too many people take it as dogma and do not question the area they have explored. My advice is to take what you can from the discipline well keeping in mind that every piece you take comes with a centuries long dialogue.