External resale Marriott Weeks are not eligible for enrollment in the Destination Club Points program, which is the only internal Marriott exchange program,* so your exchanges would have to be external. You could use any exchange company but Marriott is affiliated with II so the overwhelming majority of external exchanges are processed through II. There is a reduced exchange fee if you deposit a Marriott Week and include only Marriott properties in your search parameters - even if you include non-Marriotts in the search and confirm into a Marriott, you'll be charged the higher exchange fee. I'm sure someone more familiar with II will be happy to tell you the current exchange fees.
Owner deposits to II do have a Marriott preference attached to them in that they're only available to other Marriott owners for the first 21-24 days. There are exceptions to the preference rules - the time period is reduced to 3 days if the deposit is within 60 days of check-in; and, some deposits made by Marriott of Marriott-held inventory do not carry the preference.
Availability of specific inventory is never guaranteed - unless you're talking about Fixed Weeks - either as owners or exchangers. As should be expected the highest-demand Weeks throughout the system can be difficult to get in exchange because Owners use them (or rent them privately.) I don't know if Marriott/II exchanging is "better or worse" than it used to be, but I wouldn't count on it if your aim is to get high-demand intervals in exchange for low-demand Weeks.
*edited to add - DOH! As TheTimeTraveler said, there's an internal exchange system among a few resorts in The Florida Club; see
this document from TUGger dioxide.
Moderator Note: I'll be merging this thread with your similar thread from a month ago, to make it easier for you to review all of the current information.