11 yo top loading Sears washing machine prob

You basicaly have to decide if its worth fixing considering how old it is, or perhaps better off getting a new one.

The oil is getting onto the clothes? Where is it coming from? The only thing I know of that has oil in it is the gearbox which is below the motor on my direct-drive Kenmore.

Leaking oil means transmission. You can net search about this. My gut hunch is that it's more expensive to repair. Shears repair tends to be more expensive than any other company. So, try some local (to you) appliance repair shops, see what they say.

I see you're posting from Vermont:

Here are some used machines:

$50 - Poultney

I've even seen free washers on Craig's List that needed a new motor or clutch (supposedly). I bought a used motor for mine from a used appliance place and it was only $30. I suspect the trans/gearbox would not be a whole lot more (used). OTOH I found it a lot cheaper to get the coupler and dogs on eBay than from the appliance store.