Republicans wrested the governor's seat from Democrats in Missouri and Washington yesterday and were leading in a handful of other states as they made a strong bid to end a lopsided Democratic domination of statehouses that has lasted a decade.

In Washington, Republican John Spellman led Democratic state senator James A. McDermott, who had defeated controversial Democratic Gov. Dixy Lee Ray in the primary.

In Missouri, former governor Christopher S. (Kit) Bond held a steady lead over incumbent Democratic Gov. Joseph P. Teasdale, who had outsted Bond four years ago. Republicans also appeared to have a chance to oust Democratic incumbents in Arkansas and North Dakota.

In addition, Republicans easily held onto the only three governorships they had at stake yesterday, as incumbents Pierre (Pete) du Pont of Delaware and Richard Snelling of Vermont won reelection handily, and Lt. Gov. Robert Orr kept the Indianna statehouse in GOP hands. Republicans mounted a strong bid to cut into the big edge, now 31 to 19, that Democrats had held for a decade.

Democrats reelected incumbents John D (Jay) Rockefeller IV in West Virginia, James Hunt in North Carolina and J. Joseph Garrahy in Rhode Island. In a hard-fought campaign in New Hampshire, Democratic incumbent Hugh Gallen was beating former governor Meldrim Thomason Jr. Coupled with 21 holdovers, these four victories would give the Democrats a minimum of 25 of the nation's 50 governorships.

Democrats were favored to retain two other governorships they held -- in Utah and Montana, and they were leading in both races with about one third of the votes counted.

In Arkansas, an upset appeared possible as Little Rock banker Frank D. White was edging ahead of Democratic Gov. Bill Clinton late in the evening.

In North Dakota, incumbent Democratic Gov. Arthur A. Link was trailing challenger Alan I. Olson.

The Republican National Committee claimed that the GOP had gained control of the Ohio Senate and the Illinois House as results for state legislative races slowly trickled in.

State legislative races were especially hard-fought yesterday because of their role in the redistricting of Congress next year. For years, Democrats have enjoyed big margins in the legislatures. However, once the results of this year's census are known, it will be the state legislatures that will redistrict the congressional seats.

With big shifts in population over the past decade, several large northern Frost Belt states like New York are expected to lose congressional seats to the fast-growing South and West. At least 14 seats are expected to be lost to the Sun Belt. The party that dominates a state legislature will then redraft the boundaries of whatever seats that state has kept or gained. Normally, the party that is in control will draft boundaries to give itself the maximum number of seats.

Democrats have had strong control over most governorships and state legislatures for a decade and more. Going into the 1970 elections the GOP had 32 governorships, but they dropped to 21 that year and the Democrats have had a majority of governors ever since, with the GOP plummeting to only 12 in 1976.

In the state legislatures the Republicans have trailed even more. Democrats have controlled a majority of the legislatures for the past dozen years.

In the past few years, Republicans have made a determined effort to boost their governorships and state representation, with a measure of success. Spending about $2.7 million in the state legislature races in 1978, they managed to double the number of legislatures in which they controlled both houses from six to 12. Democrats, however, had majorites in 30 legislatures going into yesterday's election, with the remainder split or nonpartisan.