More than likely the compression release mechanism isn't working correctly (stuck in the running position). When the engine is moving slow, there's a little mechanism which allows opens a valve and allows compression buildup to be released. This allows you to spin the engine fast enough to get it started. As the engine gets up to speed, the mechanism, due to inertial forces, is moved out of the way and the compression is allowed to build as it is supposed to so the engine can run correctly.
Here is a short video on the B&S engine's compression release.
It is one of two things happening. Either there is buildup around the mechanism which isn't allowing it to return to it's starting position, or the return spring has become weak/broken and isn't doing its job. You could try an engine flush to see if it'd help (as per the first scenario), but more then likely you'll need to take the engine apart to fix it. If you are mechanically inclined, you should be able to fix this at home. And knowing the price of replacement engines, it would make economical sense to fix it rather then replace it.