In my opinion the old sod should should all be removed, preferably with a flat shovel blade under the sod. If you have a large area you can rent a machine to do this. I would also haul it all away, not try to compost it. Your soil has not seen the light in many years and will be badly compacted without any humus in it. For mine I worked in about 3 inches of finished compost with a rototiller, then leveled and rolled it with weighted lawn roller. I bought fescue sod directly from the grower and rolled it in. It requires less than half as much water as my old lawn. Al

When you say crabgrass, do you mean the annual grass or do you mean the stuff that grows long stolons, that is sometimes called Bermuda grass? If your lawn had Bermuda grass in it, it will grow right up through any sod you lay down.
Renee

Annual crabgrass is also a guaranteed return visitor as soon as it becomes early spring again. There are no shortcuts to establishing a newly sodded lawn that will grow deeper roots/become more drought tolerant, resist weeds and grow well. Get over the idea that you can take your idea of shortcuts and it will work out well. If you do have a perennial rhizome type grass like St Augustine/Bermuda grass or Kikuyu grass, you'll be better off treating with herbicides several times before using a sod cutter to remove it. Do a search on the topic, all the best strategies pro and con have been well discussed here many times in the past.

Unless you make the effort to properly prepare the area, adding new sod will just be a complete waste of money.