Q: Weird sound and shaking as well as lower gas mileage and poor acceleration

It sounds like you may have a clogged catalytic converter. As you may already know, the catalytic converter converts toxic exhaust gases into less harmful pollutants that are suitable to be expelled into the atmosphere. When this is clogged, this causes these gases to be backed up into the exhaust system not allowing the engine to breathe properly (think of a backed up sink). When this happens, the engine is forced to work much harder to expel these gases. This often results in the engine bogging down, hesitating, shaking and causing very poor fuel economy. This should also cause a check engine light to come on as the O2 sensors should be reading the incorrect amounts of oxygen in the exhaust gases if this is your problem. I would recommend having an expert from YourMechanic come to your location to diagnose and replace your catalytic converter.