It sounds like a problem with one of the relays on your control/logic board. The relay is sticking in position until all power is removed then it sticks again once power is put through it.
Unfortunately, the relays cannot be replaced individually as they are permanently fixed to the board. Considering the cost of the logic board you may want to have a professional confirm the diagnosis in person or may consider replacing the opener completely but those decisions are not ones we can make for you.You can get a better idea of the part costs by checking under your model number at Sears PartsDirect and determine what service options are available in your area and the related costs by calling 1-800-4MyHome.Sears.Com and your Sears Store will have a selection of replacement openers in various cost ranges depending on which features you want and you will also want to consider potential cost or time needed to install any replacement opener.