You might want to Google how to wash wool to understand better what does and does not make wool shrink.
Personally, I would *not* take this quilt to a dry cleaner. Do you have access to a top-loading washing machine large enough to handle this quilt? If so, here's what I would do. Fill the washing machine with cold water and a gentle detergent (Orvus paste is good and available from many quilt shops; also available at Fleet Farm and other places that sell horse shampoo). Stop the machine! Add the quilt, and gently push down on it (hand agitation) to run the soapy water through. Advance the washing machine to the spin cycle (do *NOT* allow machine agitation) and spin out the soapy water. Fill with rinse water. If the quilt is still dirty -- and it is very likely to be, after 37 years -- add some diluted Orvus and hand agitate again. Advance machine to spin cycle. Repeat this until you are satisfied all the dirt is out, then do the same process with cold rinse water several times.
To dry, lay a large flat sheet on the grass outside in the shade. Lay the quilt on top of the sheet, gently "blocking" into its rectangular shape. Cover with another large sheet (important for keeping bird droppings off the quilt!) and anchor the edges with some rocks. You will probably need to flip the quilt at least once, possibly several times, in order to get it dried thoroughly. When it is just damp instead of soaking wet, you can block it a little more if you like. You can place fans on the quilt to speed the drying process. (And you can dry it flat inside the house instead of outside if you have a flat surface big enough.)
Here's the thing with wool. If you look at wool fibers under the microscope, the fibers have tiny hooks. These are what cause wool to shrink and "felt". The hooks interlock when in the presence of hot water, detergent (makes the water slippery), and agitation. As long as you use cool water and do not machine agitate, the hooks will not interlock. You do not want to use a dryer because the tumbling in the presence of heat will encourage those hooks to interlock. You do not want to hang it on clotheslines because water adds a lot of weight to the quilt, and that weight puts a lot of stress on the threads holding the quilt together. Best to dry it flat.
Your biggest problem will probably be finding a top-loading washing machine large enough to handle the bulk and weight of this quilt. Although you could do it by hand without a washing machine, the weight is going to be monstrous once all that fabric is wet (water is heavy), making it very difficult to manipulate and carry.
Hope this helps!