Currently on my grace period in the US. Is it possible to apply for an ESTA in Canada (drive through the border) and get a few months more in the US? Would there be any issues in entering Canada and clarifying I’ve left with my J1 visa. I have multiple nationalities, should I stick with the one of the expired Visa?
To enter the US under the visa waiver program by land, you don't actually need ESTA. Also, there's no need to "clarify you've left with your J-1 visa" because once you've left the US your J-1 status ends, and because you're in your grace period you are not eligible to re-enter in J-1 status.
But to answer your real question, yes, it is possible. There is a risk that you might not be admitted to the US, however.
You can also apply to remain in the US through a change of nonimmigrant status to B-2. This option is not available to J-1 nonimmigrants in a couple of cases, however:
If you are an international exchange visitor (J-1 nonimmigrant), you may not apply to change your nonimmigrant status if:
You were admitted to the United States to receive graduate medical training, unless you receive a special waiver, or
You are an exchange visitor and are required to meet the foreign residence requirement, unless you receive a waiver.
Switching nationalities won't help you at all, and might cause the authorities to doubt your intentions.