Horsh answered for the Russian side of the station very. But the question was more about the US side of the station.
The US side of the station does need more ports. It looks like a fair bit of reconfiguration will be performed of the current modules to free up spots.
NASA desires to have two of each type of port available as a backup. Thus they would like two CBM ports available when cargo flights arrive, in case of an issue with one port, to allow berthing regardless. The same is true for the docking ports for manned flights.
A great article discussing the reconfiguration explains how this will be handled.
The PMM (Permanent Multipurpose Module) which is currently on the earth facing (nadir) side of Node 1 will be moved to the forward port of Node 3. (Which would put it parallel to the US Lab module). Interestingly, the Bigelow Expandable Activity Module (BEAM) will be on the aft port of Node 3 (facing the Russian segment).

That is, the PMM, facing down in thisthe photo:

above, will move onto Node 3 and you can seesuch that it beingis parallel to the US Lab in this photo, and you(photo below). You can also see in the image below the 2 PMAs on Node 2 (one facing forward, one facing upwards):

That will leave the Node 1 nadir, Node 2 (where the PMA is currently docked facing forward) nadir (earth facing) and zenith (space facing) CBM ports available.
Then PMA-3 which is currently unusable on Node 3, will be moved to Node 2 Zenith.
Thus the ports available will be:
#Docking ports
Node 2 (where the shuttle used to dock, at the front end of the station) will have a PMA facing away from Earth, and a PMA facing forward. Those will be available for manned crew vehicles, which the current schedule only ever has a single one at station at a time. (Indirect handoff, instead of perhaps both docked at once, and one leaving after a week for the handoff). These PMA's will be upgraded with a new docking ring to meet the NDS (Nasa Docking System) standard that all currently planned manned vehicles will be using. (Amusingly, Dragon will be delivering the NDS adapter, and the BEAM module to the station).
#Berthing Ports
Then the Node 2 nadir and Node 1 nadir will be available as CBM ports for berthing cargo vehicles. This is preferred over using the nadir and zenith ports on Node 2, as using the top and bottom of the same module would require the CanadaArm2 to berth only to the nadir side (Capture box is below the station, not above it) disconnect, change its grip on the station to then reposition the vehicle from nadir port to the zenith port. This would then require that the vehicles depart in order as well, restricting access more than NASA would like.
Currently the lack of ports is being handled by restricting access to the station to one vehicle at a time. The reconfiguration will allow two cargo vehicles and two crew vehicles at once, but may not be used in the crew configuration.
But the restriction on access is already starting to cause issues as Dragon, Cygnus, and HTV all share the CBM on Node 2 nadir right now for cargo, and thus scheduling becomes a complex dance with many dependencies and a delay in one flight ripples down the manifest for years.