1. There is no set screw.

Fill the tub about halfway with very hot water to soften the plastic a little, drain the tub, then put your fingers under the rim of the agitator and pull it straight up. It will resist, then suddenly fly up and hit you in the chin. This is all normal. It will have been in place for years and may be very stuck. (They are retained by a little rubber ring in a groove of the agitator shaft.)

2. The black marks may be grease or oil getting past a worn out agitator shaft seal, but if you wash in cold water, it is more likely to be built up grunge - a mixture of detergent residue, dirt, body oils and especially fabric softener, if you use it. After a while it turns nasty and resembles black grease, bits of it break off hidden surfaces and get into your wash.

Several very hot washes with plenty of detergent will fix that. Cold water washing is good for your power bills but hard on washing machines.

Good luck with it.

For people who don't know these machines, the splined agitator shaft is like some Maytags, but the transmission is quite different - a cast iron frame inside a plastic case, it looks like a tupperware canister full of oil. These are small machines - the original was the Minimatic, with a porcelain steel inner basket; later, they made it taller, using the same basket from the Minimatic, with a plastic extension on top to make the tub taller. This was the Delta 10. This taller tub distinguishes the Delta from the Minimatic. the original Delta was called Delta 10,  and had small rocker switches for temp and drip dry options; this later Delta S changed to selecting temperature on the timer dial and no drip dry option, no rocker switches.

Hi Mike

Yours is a much newer machine, a whole different animal really.

Simpson Minimatic, Delta 10 were 1970s. Minimatic was fixed wash level, Delta 10 had variable water level and taller tub for larger capacity. They were facelifted in the 1980s, Minimatic and Delta 10 names continued with new fascia; Minimatic gained variable water level. at some stage Delta S came out as a simpler, slightly cheaper alternative to Delta 10. (simpler control panel.) These machines were like a scaled down version of the big Simpson, which was very similar to the US Maytag. I'm not sure if any licensing of Maytag technology was involved, or if Simpson just looked hard at a Maytag before designing the Simpson, but the agitator of the big Simpson looked just like a Maytag.  The Mini and Delta were quite durable but noisy and not very good spinning - slowish spin and small tub diameter. Not terrible, but not great either.

Over time the Japanese GE, Hitachi, Toshiba  and Sanyo machines (including under "Aussie" brands such as Whirlpool, Wilkins Servis and Philips) started eating sales of the Mini and Delta, so Hoover and Simpson responded with their own medium size washers that were more or less copies of the Japanese machines - Hoover brought out the Premier and Simpson brought out the Genesis. These were two quite different machines, but both worked the same as the Jap clones. Hoover had a 7:1 reduction, nylon gears, and a big flat agitator; Simpson has 4 or 5:1 reduction (I forget), steel gears and a smaller, more conventional agitator. Both had early teething problems but became very popular and eventually their whole top-loading ranges were derivatives of these two machines. Simpson made Simpson and Westinghouse branded versions (And I think briefly Kelvinator, too??) ; Hoover made both Hoover and Admiral branded versions for a while.

Your Delta would be the next size up from the original Genesis, from vague memory. The "Genesis-era" generation of Deltas, like yours, are just called "Delta", not Delta 10 or Delta S.

If anyone else wants to add detail or correct me here, feel free. These are my somewhat vague memories. Nathan or Leon in particular might know more detail.