Problem solved!
I just filtered 2 quarts of water with only gentle squeezing at a rate of about 1.5 minutes per quart. That's about 5 times faster than the flow rate that I was getting. Ha ha, Jerry, my filter is now faster than yours.
I think running hot water trhough the filter was the key. Things improved immediately. Vinegar may have helped too but I'm not sure.
Thanks for all the help. You saved me a trip to REI.
PS, I called Sawyer to let them know their suggestions worked. She added that after backflushing they suggest running a couple of liters of clean water through the filter (in the normal, non-backflush direction as shown by the flow arrow). I think this is to flush out any dirty water that was sent upstream during the backflushing.
She also said periodic use of hot water directly from the faucet would not be a problem for the filter. I found the water faucet easier to use than the syringe. She said doing so immediately after use might prevent deposits from getting attached to the filter fibers. Removing deposits before they dry on is much easier than doing so after they have dried on. (this is a little confusing if the deposits are coming from the tap water but we'll see) She also strongly recommended following the bleaching protocol prior to storing the filter.