Now is good for both transplanting and dividing.

Method for division is to give the grass it's haircut, then dig it out, taking as much rootball as you can. Use a very sharp tool like a spade, axe, etc. to cut the crown and rootball into as many parts as you want.

Very large plants can be divided in place, with the same general technique. Use the sharp spade to halve/quarter the plant in place, and dig out each division.

It's best to prep the new location(s) in advance, so you can plant the divisions immediately.

Now's the time in CA. Now or soon for NJ.

If you've got a big grass, I have found it is best to dig up a rootball and divide from the side with a spade, because sometimes it is impossible to penetrate the clump from the top.

I have moved miscanthus grass from the first signs of greening to 30" high. The best results were immediately after the first few green shoots started to grow. In moving mature clumps of miscanthus, I dug 12" to 18" outside the diameter of the root clump and 24" to 30" deep. I wrestled and rolled the huge clumps onto a small utility wagon and pulled them to their new location. After transplanting, I kept them well watered over the summer. Note that the moved grasses did NOT regain their prior mature height until the following summer. I believe the quick recovery was in part due to the preservation to a large portion of the roots. When replanting, I like to mix peat moss with the soil and add a little slow release fertilizer and triple phosphate. Note that large clumps of freshly dug miscanthus are extremely heavy, which require careful handling to avoid muscle and back injury.

I must agree with Deep_Roots 5a (what's with the twin names??)
Moving 'can' be done at any time of the year if you take care & maintain a large root ball. The 'best' time with the least amount of shock, is when the grass is showing a small amount of growth (new leaf shoots peaking through)in early spring.
Division, (IMHO) on warm season grasses is 'best' done in early spring before the leaves start to unfurl ~ this results in the least amount of shock to the plant. Wait until they have shown some signs of growth ~ no point dividing out & trying to grow dead portions :o).

Cool season grasses can be divided pretty much any time but early spring and fall (like most of your other perennials) is the best time.