Murray mower with B&S 500e engine running strange
For those symptoms, I'd initially suggest that the fuel tank and fuel-line need a flushing and cleaning. The tilting back remedy typically means the fuel tank's screen filter is mostly clogged...assuming it has one or some type of filter.

I don't know the specific engine, but very related to the above is a fuel-bowl attached to the carburetor and most especially the Jet-Bolt that attaches them together.

The fuel-bowl needs to be clean and the Jet-bolt needs to be poked clean with a pin and then blown or flushed clear.