Q: How should I flush my engine oil system?
A: To flush the engine, before the oil change drain a quart of oil out and replace it with a quart of transmission fluid (ATF), and drive it without stressing it for a couple of hours, maybe longer. Drain the oil and change the filter.
Many people recommend Seafoam for flushing, but I don't believe in anything but tranny fluid. ATF is very close to motor oil in composition, but has a really high detergent content. More than one old mechanic recommended tranny fluid to me and I trust them explicitly. They know more about engines than I ever will.
I would seriously recommend against using solvent type flushes. Solvent flushes are only good for varnish, not sludge or other hard deposits. They can dissolve sticky deposits which hold other deposits in place. If the released deposits are not soluble in the solvent, they will be loose to cause trouble by blocking oil passages. This can kill your engine really fast. This is the advantage of using transmission fluid. It is designed to suspend deposits with a detergent additive which should break up particles before they can stick somewhere bad.
If this doesn't work, I would try
AutoRX which is specifically designed to dissolve sludge. You use it as directed on the bottle. AutoRX is not a solvent, but is a type of oil which softens sludge and allows the oil to break it up. Read about it at the above link.