Thank you, chillfactr and powrslave, for the replies. Much appreciated.
I was trying difficulty of 128 before but went and tried the 32 as suggested. What I never realized is that there seems to be two pools behind this software. First pool is Pool #0 and the second is Pool #1. The pool "management strategy" is set by default to Failover so that would indicate two pools, at least. There is also Round Robin and other management strategies to choose from.
What is happening now is that when I use the --suggest-diff 32, it seems to be modifying Pool #1. Pool #0 still sits at difficulty 10,000. And the Pool #0 seems to be the default because when I use verbose display mode it still says 10,000 difficulty. I can manually force the software to use Pool #1 with the lower difficulty, which I did. Now it's mining with the difficulty of 32 (well, it raises it to 72 automatically). Still "Share above target" continually.
What is odd, though, is that even though it prints "Share above target" using Pool Management -> Information I see this:
cgminer version 4.11.1 - Started: [2021-03-15 05:56:56.210]
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(5s):149.1G (1m):137.6G (5m):134.9G (15m):130.1G (avg):134.0Gh/s
A:106510 R:144 HW:23 WU:1872.7/m
Connected to solo.ckpool.org diff 72 with stratum as user BTC_address_removed.Worker1
Block: 801f1356... Diff:21.4T Started: [06:47:35.338] Best share: 83.7K
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\[U\]SB management \[P\]ool management \[S\]ettings \[D\]isplay options \[Q\]uit
0: GSH 10038528: BM1387:02+ 200.00MHz T:200 P:200 (47:94) | 99.8% WU: 98% | 46.41G / 44.54Gh/s WU:622.3/m
1: GSH 10038553: BM1387:02+ 200.00MHz T:200 P:200 (47:94) | 97.8% WU: 98% | 41.45G / 44.51Gh/s WU:621.9/m
2: GSH 10038573: BM1387:02+ 200.00MHz T:200 P:200 (47:94) | 100% WU: 99% | 59.06G / 44.98Gh/s WU:628.5/m
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Share submissions: 1311
Accepted shares: 1309
Rejected shares: 2
Accepted difficulty shares: 89848
Rejected difficulty shares: 144
Items worked on: 178081
Discarded work due to new blocks: 28618
Stale submissions discarded due to new blocks: 0
Unable to get work from server occasions: 0
Submitting work remotely delay occasions: 0
It seems to be mining anyway?
Unless I am misunderstanding, it does seem to be mining.
Please correct me if I am wrong or if the results don't look right for the hashrate. I am very new to these Newpacs and to cgminer and I appreciate the advice and expertise.