More details would be very much helpful, such as ... what brand & model is the washer? ... Does the problem happen on all cycles (Regular/high speed & Delicate/low speed)? ... Does the machine sound and run normally during operation, or is the motor obviously turning rapidly on/off with the flickering?

The model is a GE WBSE3120. Yes, it happens on all cycles. The motor doesn't seem to be turning on and off during flickering but it does seem like it's struggling a bit.

How old is the house? sometime I see where the neutral line at the fuse box is loose and any higher power motor causes the lights to dim when it runs. Some older houses are set up for 50 or 100 amp service and this may not be enough with all the modern conveniences of life plugged in.
Since you tried the washer on various circuits and the washer doesn't seem to be struggling I would suspect the house wiring. If your house is a Mid 70's vintage you may have aluminum wiring and this could be part of the issue

I'm not familiar with these GE machines (which have a bad reputation), but found some info advising that a binding motor/clutch can cause this problem, and also possibly the control board.

Your model number should have four more characters ... a letter B, a number 0 or 1 or 2 or 3, and WW or CC (which likely is just a color code). The number is a series or engineering change code and is relevant to selection of correct replacement parts.

I have copper wiring and 2oo amp service and a relatively new panel. The house was built in 1969 but like I said the power has been updated. I called a local repair company. He thinks that if the product is running through the cycles that it may be my power company dropping the voltage a bit over the summer months. He told me not to rush and do anything and see if it improves after summer. I thank everyone for their help and suggestions and I still welcome more ideas if anyone has any.