I'm looking for a few select low-level Legendary items to extract their special abilities using Kanai's Cube. How do I do that with only level 70 characters? Do I need to start a new character?
The level of the "low-level" Legendary items is the minimum level those items will drop at.
While at level 70 the list of possible drops is bigger, you can still get all the drops you need and it is definitely going to be faster to do so on a higher difficulty at lv70, rather than a lower difficulty (with low drop rates) at a lower level while also trying to avoid leveling up.
If you're looking for specific low-level legendary items in order to extract their power, it can be beneficial to create a new character of the class you need and get them to the minimum required level for the item you want. Then either gamble or upgrade rares to legendaries. This is an extremely efficient way of finding very low-level legendaries like Lut Socks, because the item pool of possible legendaries is very small.