The unexpected death of someone close to you is a tough situation; more than that, it is one of the archetypical tough situations throughout human history. The obvious -- but not easy -- general answer is that you need to either become more self-reliant, find other people to satisfy the needs and desires that were being met by the departed party, or some of both: some of both sounds healthy to me.
It sounds like your department is trying to work with you to meet your needs, which is great. My first comment is that's an ongoing process, not a one-time decision or fix. How are you supposed to know right now what accommodations are needed or optimal? You can't. You should identify specific departmental personnel and make clear that you will be checking in with them periodically about the situation.
I have heard some people advise on transferring institutions at this point, though my department does not recommend that.
Whether they recommend it or not, you can leave at any time, and -- with a deceased thesis advisor -- an absolutely clear conscience. So if you know of a faculty member at another institution that you think would be an ideal advisor, feel free to explore that right away. It sounds to me like you don't, and I'll continue under that assumption.
Your department's recommendation may well be reasonable....but do you understand why they're making it? I would hope that by saying this to you they have some degree of planning for your successful completion of a PhD. You say a little bit about this, but not enough: they offered for you to work with postdocs, but you say that there are no faculty in your field. That's a bit ambiguous: are these postdocs in your field, enough to help you finish your thesis? I'm guessing they must be at least close, because if not you could get a tenure-track faculty member who is not in your field.
I think you should talk to these postdocs and get a sense of whether they could help you finish your thesis. If so, that sounds like a good option for your situation: basically they become your new thesis advisor, and the fact that they are not tenure-track at that institution is not so directly relevant, so long as they will stay there until you graduate. (Look into that!)
For the majority of students in pure mathematics that are not in the home stretch of their program, being without a designated advisor at their home institution would leave them terribly adrift: a few plane trips here and there will not cut it. Recently I was on the committee of a student whose advisor left the department soon after she started working in earnest on her thesis topic. They kept in close contact through weekly skype appointments and a faculty member in a related field stepped up to become her official thesis advisor. I watched this happen and would say that it set her back a full year. In the end she did graduate, get a good visiting position, and now seems to be doing extremely well. The moral is that ultimately it is all about the student: this was a real test of her character, and she passed with flying colors, but gosh it was hard, even hard for me to watch. (And I must tip my hat to my departed colleague as well: he was still closely involved with the student the whole time, infinitely more so than he was obliged to be.)
I would say that you in particular sound like you really need a new advisor. Don't stick around at your home institution without someone stepping up to that role. (And certainly don't wait around for them to hire someone else! Unless they can tell you right now who they want to hire next year, in which case you should ask to spend the intervening time at the present institution of this person.)
You ask about how academia works without having an advisor to mentor you post-PhD. To be frank: by asking for a quantification of the marginal difficulty of the deceased advisor and in other ways as well, your question is telegraphing that you have an uncertain commitment to a post-PhD academic career.
An academic career in pure mathematics is hard for everyone, and thinking that the key to success is close post-PhD contact with your advisor sounds closer to wrong than right to me. I will admit that I had a fairly extreme situation: my former PhD advisor is one of the great mathematical luminaries of our time. He gave me the help and guidance I needed as a PhD student. But I have never collaborated with him, and since graduation I haven't asked him any mathematical questions. We remain on perfectly good terms (I have his phone number) and have professional interactions like two mathematicians in the same field, but once I got my PhD he became my former advisor (though that is still a critical role: e.g. it involved writing a lot of letters on my behalf!), if you take my meaning. One of my oldest friends got a math PhD. A few years later, his advisor died in a tragic bicycle accident. So that meant the end of the mentorship, and I remember discussing the awkardness of the situation with him for several years after. My friend is now the chair of the math department at his university. Again: ultimately it is all about the student.
To sum up: as others have said, this may well be a traumatic situation for you. Take it as such, give yourself some time to grieve and to let things sink in. While you're doing that, involve your department in your plans to complete your PhD there or elsewhere and to find at least one new mentor. When you do find a new mentor, see if you can regain mojo you might have lost during this tough setback. But if you really think of mathematical research as something that you need someone else's guidance and energy to pull off, I would recommend that you complete your PhD and then look elsewhere for a career: either a teaching-intensive job or something else entirely. The main benefit of a research career in mathematics is that mathematics is frickin' awesome and spending your life doing it makes you a rock star. There are other benefits, but none that stack up against the sizable costs.
Added: You speak as if you might not even finish your PhD. For someone who's four years into a math PhD program and already has enough results for a paper, I think not finishing the degree ought not even to be an option on the table. The more I think about it, the more I wonder if your natural emotional reaction to your advisor's death is playing a role in your thinking. Trust me on this: everyone wants you to finish your PhD. It's a cliche to say "That's what your advisor would have wanted", but it's probably true, right?