Unfortunately it does take quite a bit of solvent to properly clean all spray equipment. If you think of the quantity of water that would typically be run through after spraying latex it makes sense that a similar quantity would be needed of whatever other solvent is required for another paint type.
This 'wasted' solvent is one of the under-advertised costs of spraying generally.
All of the solvent used for cleaning doesn't have to go to waste however. If you're spraying paints in particular, the dirtied solvent should be retained, and after some days or longer most or all of the pigment will have settled out of suspension into a dirty layer at the bottom of the jar:
The apparently clean spirit on top can be decanted to a fresh container. This can sometimes be done effectively by just pouring it away, but a large syringe or turkey baster may be preferred to do the job more cleanly.
The spirit layer above the settled gunk may look perfectly clean but there will be some dissolved binder/carrier from the paint in solution so it's slightly contaminated and should never be returned to the original container.
The slightly dirty solvent can still be re-used a number of times, but there comes a point when the contamination becomes too great and it should then be safely disposed of (otherwise it will leave a sticky residue behind on anything it is used to clean, as it is in effect very very dilute varnish by this point).
I've been doing this with Laquer thinner but feel that I'm using too much. It seems to take almost a small quart to clean my machine.
Should I approach oil based differently, and how do you do it?
Well lacquer thinner would be overkill for oil-based paints normally, mineral spirits (UK, white spirit) or turpentine would be the solvent of choice for any and all paints of this type.
Some people do use lacquer thinner as a final flushing rinse after cleaning the equipment with spirits or turpentine but generally the only reason you'd need to resort to something like lacquer thinner or acetone is to clean equipment where the oil-based material has had a chance to fully dry, whereupon it will have become insoluble in spirits or turps.