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The 737 MAX was introduced relatively recently. I would have assumed they were replacing at least some older 737 models in operation with the major airlines that took them (eg Southwest, American, United, in the US). Did those airlines immediately sell off their older planes as the new ones joined their fleet? I see reports of huge flight cancellations from lack of the new aircraft. Since they have common type ratings with the earlier 737, what happened to the older planes that none of them can be pulled from storage or dry leased and put back into service? Surely this grounding is long enough to be worth the transient expense?

It really isn't 1 new aircraft = 1 mothballed aircraft. Most of the 737 MAX aircraft were fleet expansions (for US carriers). The older aircraft get retired when they aren't economical to fly anymore, and even then a retired aircraft is often sold either to a scrap yard or to a lower budget operator. In order to bring an aircraft back into service they may have to go through C or D checks before they can be considered airworthy again (because old aircraft are often retired at a D-Check when it needs a major check).

Airlines need to schedule C/D checks out months in advance, each aircraft costing millions to go through the check and a month or more of downtime. There is only limited space available to do these checks, so even if they wanted to bring 10 aircraft back into service, it may take 6 months and cost $20-40 million. It's cheaper to cancel the flights.

The main problem is in the United States, the 737 Classics are banned beginning December 2017, due to fuel tank interting requirements. In particular, Southwest's Classic replacement schedule was driven by this, their last flight was in October 2017. They have a number of parked Classics, but they can't be operated in the US. On the other hand, the 737NG is new enough that there isn't a significant excess fleet.

Due to the TWA Flight 800 explosion, FAA required retrofit of fuel tank inerting on most aircraft with center tanks. This system purges some center tank oxygen with nitrogen, in order to reduce the risk of explosion if there's a spark. EASA did not require retrofit, and the MD-80 is exempt (no heat sources by its tank).

While a retrofit was developed for the 737 Classic, the cost of the system, coupled with the age of the aircraft, and the 737 skin issues, meant that all the big US carriers chose to retire remaining 737 Classics as of the deadline, rather than retrofit.