Is there a way to trade equipment within my party that I am missing, or am I stuck with it staying the character who picks it up even though another character might find it more useful?
Hey @hellrazoromega, and welcome to the forums!
Yes, you can change equipment within the party. 
When you’re inventory, just mouse over whatever equipment you want to unequip and hit the right mouse button (X if you’re using an Xbox Controller or Square if you’re using a PS4 controller). That will unequip the item in question.
Then you just switch to a different character and give the item to them instead! Easy peasy.