NEW YORK—Corn and soybean prices pushed deeper into record territory before easing Friday as rain again soaked the Midwest and traders locked in profits ahead of a planting report next week.

The early gains followed a sharp run-up in commodity prices over the previous two days, and came as oil futures touched a new high just shy of $143 a barrel.

Strong downpours continued to hammer key growing regions in a number of Midwestern states, including Iowa, Illinois and Indiana. In Missouri, an earthen levee along the Mississippi River broke, raising fears of flash floods in ...