These 'artifacts' are a well known problem, and are usually the result of polygons crossing the antimeridian (180 degrees e/w)
The go-to fix for this is usally ogr2ogr with the wrapdateline option.
But that won't help you. In your case, you're using an offset around -156. This means that any feature crossing 24E meridian (-156+180 = 24) is giving you problems.
To fix this, I removed a thin strip either side of 24E.
I started with Natural Earth data, and left off projection (for now), and just used WGS84.
To draw the 24E meridian, I used the QuickWKT plugin and added the following as a new layer...
LINESTRING (24 -90,24 90)
That draws a single line along the length of the 24E meridian.
Next, I manually digitised a polygon scratch layer, adding two polygons, one to each side of the line, and a hemisphere in size, but hugging the line as close as possible. (Note the quality of the line drawing here...)

You should probably do that with the QuickWKT plugin too, to get more precision - it involves more typing and I wanted a quick test :)
Next, I used clip to clip my original shapefile to the layer with the two polygons. This cuts out a thin strip around the 24E meridian...

finally, I applied OTF projection using your custom CRS - and the fixed result.

+lon_wrapoption. – AndreJ May 19 '17 at 05:56