HI Leila:
Welcome to the iris forum.
I'll try to help with some of your questions. Sorry to be the bearer of bad news, but this is not the time of year to replant iris in NJ. Should be done before 9/15. However, I know people in NJ who plant this time of year and still get the plants to live. If you have tough older cultivars they may still make it. You can do one of two things:
Replant after preparing a bed for the iris ASAP, or place in pots and put in the garage for the winter. You should divide the iris into sections where there are one or several 'eyes' where the fans are starting to emerge. A 'Y' is OK . A large network of 4 or more 'eyes' should be broken up. Just pull apart. Smaller pieces just replant as is, but they will have a hard time surviving since they have not had a chance to develop a root system. If you can, plant some in the ground - shallow so the roots are buried but the rhizome is partially submerged in the soil, and plant some in pots to put in the garage (or other cool dry area) You can then plant in the ground in the Spring. I don't know what the whitish soft lumps in the roots are, sounds like some kind of fungus. A rhizome should be firm like a potato, and the rhizome and some string like roots should be all that an iris plant should posess underground. I would remove any soft lumps like you describe before planting.
You may suffer heaving in the winter, so you will have to check them and put a rock on top of them to help prevent it.
You could also mulch heavily with straw, evergreen boughs, or Remay horticultural cloth to give them more protection since they hav enot had a chance to establish themselves before the frosts come.
If they survive, do not expect bloom in the Spring, or minimal bloom at best, since you missed the optimal replanting season.
Good luck,
Laetitia