Must all my DSTs be purchased and close at the same time?

No. Historically most exchanges were simultaneous but now the most common form is now a delayed or deferred exchange. A delayed or deferred exchange means you sell your property and the net proceeds go to the accomodator, and then you have 45 days in which to identify replacement property(ies) and then another 135 days to close (for a total of 180 days).