You won't save that much $$$ by leaving the siding on. IN fact, in most Municipalities, you can remove it yourself. There is info out there on how to safely do this, but first check w/your Municipality to see if there are any issues. You WILL have to pay for a dumpster to haul this stuff away. That's around $400, give or take. Be SURE that you tell the Hauler that you have hazardous waste. I suggest you order it AFTER you've done all the demolition. In fact, they can estimate the size you'll need. Now, if you want to throw MORE 'waste' into the dumpster, tell them that too. I threw away old bed mattresses, frames, tree roots, cinder block, PT wood, and so on. The reason you want to wait until you have ALL the old siding removed is because your neighbors and folks driving by will fill it before you do during the late-night hours!!!
The reasons I suggest you don't install over the existing siding follows. You'll get a wavy appearance. The old siding begins to crumble and fall down behind the new siding, making it bulge. The Installer, and for sure the Mfgr., will not guarantee the work. Some day, when you go to sell the house, you'll have to disclose the fact that the old siding is behind the vinyl.
Think of it this way. EVEN if YOU know the hazards (and non-hazards) of asbestos shingle siding, you have to think of what potential buyers will think. I'll tell you this - THey'll think the worst! It's just ignorance. Unfortunately, this ignorance will cause a potential sale to slip through your fingers. In fact, when you do the demolition, take pictures of the BEFORE and AFTER you remove the shingles. You may have to prove to potential buyers that you removed the shingles. If they see a picture or 2 of a 'bare wall', they'll be happy.
In short, don't skimp. If I were you, I'd have the old siding removed. Others may differ in their opinions but that's mine. My best to ya and hope this helps.