Yes, the checkerboard (random character) screen is normal. If there is a valid disk in the drive, the checkerboard screen will disappear immediately and you'll hear disk seeks as it boots. If the disk isn't valid (i.e. no valid boot sector found) or there is no disk, the drive light will come on for a couple seconds and then go out and the random character screen will remain. You may or may not hear head seeking, probably not as the head will already be seeked to the boot sector. If the light goes on and remains on, that may be a cabling issue (wrong connection, or dirty cable).
Also, I forget the order, but if you have two drives connected, but power up only one drive, you will get the LED on solid, and the powered up drive will spin continuously. So, either completely remove the second drive for these tests, or make sure they are both powered on, and not spinning once all drives are powered up.
To make sure your cables are correct... When looking at the Expansion interface from the rear, pin 1 is on the right side of the floppy disk connector (closest to the corner of the EI), and the cable should exit down from the edge card connector (if it's an original TRS-80 cable and not one someone made incorrectly). The pin 1 wire should connect to the bottom of the floppy edge card connector on all the drives. Note, if your cable has a striped wire (which normally means pin 1), it may not match pin 1 on the expansion interface. Druid's comments above said that the stripe is up, not down, but that would mean the stripe means pin 34, not pin 1. Don't trust the stripe. It may not be correct (I think I've seen cables with the stripe up and down). The original TRS-80 cable has keys that prevent the cable from being attached upside down to a floppy drive. If your cable doesn't have these keys, it's probably one someone made on their own. Remember these are old machines and who knows what people have done over the last 30 years.
If still no success, then clean all the contacts again, or check the cable to make sure it doesn't have a broken wire. As you said, you may have to clean the contacts inside the drives as well. But, make sure everything external is correct first.
The text on the display is rotated to the right slightly, I guess 2-3 degrees. Is that normal? Is there a user adjustment for it or does it require opening up the monitor to adjust it? I'm not sure if it's a gun alignment or the CRT has shifted internally.
It's probably the CMOS inverter(s) in your video sync circuit. I have had to fix this on two of the three Model I's I have, and at least half the Model I computers I've seen for sale have this issue (vertical or horizontal sync issues). People think it's just an adjustment issue. Nope. It's a blown or partially blown CMOS inverter IC.