A model number may help others help you.

Here are the parts
Replacement parts for KENMORE 66516354301 | AppliancePartsPros.com

The 665 denotes that it is a Whirlpool.
I think the following will be a close match to your unit but there is no 100% way for me to be sure as I have never seen a model to model cross reference for Kenmore.

This should help you delve a little further into the pump.
https://secured.whirlpool.com/Service/SrvTechAdm.nsf/2cd44500d572193285256a45004fd9d6/52521bdbd0ef763c85256cbc00520548/$FILE/8178022.pdf

Is the tubing to the upper spray arm sealing properly.

If the pump sounds a little noisier during wash it could be a low fill issue.
A common problem is the unit not filling with enough water.
This can be checked by adding a couple quarts of water after each fill during a cycle. If it now washes OK then check the fill valve.
Just be careful to not overfill it.
Often if the fill is low you will also hear a growling sound because the pump will cavitate (suck in air).
The most common cause is a dirty screen filter on the input side of the fill calve. Remove the valve from the unit and clean/check the filter.
Do not permanently remove it from the valve as it is there to keep crud out of the valve which could hold it open and cause a flood.
Add some water to the unit before restarting. Dishwasher pumps do not like to run dry.