Washer seems to get "stuck" during filling for rinse/spin cycles

I have a 20+ yo Kenmore and we noticed it was taking a long, long time to get through its cycle. It got through the wash cycle OK, but would sit at the "pause/cooldown" point while two narrow streams of water filled the tub from the rear. I didn't time it, but it just didn't want to finish unless we advanced the timer manually. I wasn't even looking at water level at the time.

So, I read through some posts and decided it was the timer, replaced it, but I noticed that it still hung up in that same spot (pause/cooldown), so I left it alone and eventually it would finish the entire cycle. I am starting to doubt that it was the timer in the first place. I haven't actually timed the cycle for the old or the new timers.

After looking through other posts, I wondered if it was a water pressure problem. I looked at the filter screen on the cold water inlet and it was perfect. I get the same amount of water dribbling into the basket whether hot or cold is selected. As it is dribbling in during "cool down" (preparing for rinse cycle), I reset the water level from large to medium load & suddenly it stopped dribbling water and started agitating. So, now I am thinking that the water flow is reduced somewhere and that's what's lengthening the fill time, the timer is fine. But it's the same on hot or cold. Could both inlet valves be the culprit?

Any help at all is appreciated.