I have two of those those machines. I would start with a thorough cleaning of the entire machine. Remove the needle plate and bobbin cover and the bobbin case. Remove the face plate, the top, and remove the machine from the case and take the bottom plate off. Clean and oil everything you see. There is a lot of moving points to be oiled.
I would suspect that the insides of the machine are dirty and dry. The stitch regulating mechanism is complex with a lot of moving pieces and it needs to be checked and cleaned too. You can do that by removing the plate at the hand wheel end and looking inside.
The 413 has one plastic gear at the top of the vertical shaft that is prone to wear and breakage. That gear drives the bobbin hook.
The feed dogs are driven by a rod affair similar to most other machines. It doesn't have a feed dog drop feature. So make sure the machine is clean and everything that moves is lubed. Tri-Flow works wonders for cranky machines.
The 413 is a noisy machine, that entire genre of Singers seem to be overly noisy.
However if the motor is overly noisy you can check that it's mounting screws are tight and that the belt is not too loose. A loose belt seems to make a lot of noise on one of those. But a too tight belt will impede performance and slow down the machine. You have to find that happy medium.
Joe