RB2, I'm new to #36 twine myself and hope someone who knows more may come along to offer you a better answer. That said, I think the stiffness of your rosary could result from the particular twine you are using rather than the way you are tying knots.
For a few years I've used a different type of cord to make knotted rosaries. My first ones were very soft and folded easily. At some point when another spool was ordered, I noticed that the cord was different, more tightly woven and it did not fray as much. My subsequent rosaries did not drape the same way and had that somewhat starched quality you describe. I questioned the supplier (a great, helpful supplier), thinking it could be due to a bonding agent, but it wasn't that. I was told that an extra step had been added to "heat-set" the cord... which makes it more durable, I believe. My guess is that your cord may have gone through a similar heat process, to set the dye color or for some other reason.
Maybe you could try using twine from another manufacturer to see if your result would be different.
"Rosary Army" would send you a sample length for the asking:
http://www.rosaryarmy.com/?page_id=52
(This is a request form for a complimentary rosary, but you could indicate that you wish to receive the small length of practice cord.)
Hope you will find the right answer.
Peace be with you.
mary
LORD, let Your mercy be on us, as we place our trust in You. ~ Psalm 33:22
Knotted Cord Rosary for Troops and for Veterans:
http://www.seatofwisdom.net/rosary.htm
http://www.seatofwisdom.net/veterans.htm