I think the issue here is that your polygons are captured using a different technique (e.g. manually captured) and scale so they don't align with the raster cells.
Here is a simple model that will guarantee the selected pixels are fully within the polygon boundary. But because the polygon does not follow the raster cell alignment this approach can miss pixels that are fully within the boundary, so if you don't mind losing a few then this approach works well.
You convert the polygon to a raster, shrink it by 1 pixel then use this to mask out the raster

Input data is raster and polygon boundary:

Result of model is:

Here is a zoomed in example of where pixels can be missed, the result of extract by mask is displayed as green pixels on top of the original raster, but all green pixels are fully within the polygon. Note some grey pixels which are fully within the polygon are not green.

The model could easily be wrapped up with an iterator to step through many polygons.
If this is not sufficient I think you'll need to convert the data into vector and intersect the data, drop out anything that is not fully within the polygon and then use that to mask out the data. This would be a slower approach due to the conversion of data but quite achievable.