While it is generally a good idea to avoid global variables there are cases where they can be useful, and your case is one of them. Do you know what a singleton object is?
A singleton is a good solution for a scenario where you need to store some data internally in your application that you have to access from several different places in your code (different units, for instance). You place the class into a unit of its own. The unit is then used by all other units that need access to the singleton.
There are different ways to implement a singleton object. The simplest one is to actually never create an instance of the class at all. Instead you use class variables and class properties to hold the data. That works well if all data you need available are simple or compiler-managed types (ordinal types, numeric types, strings). It works less well if you need more complex types, e.g. other objects. This can be handled now that Delphi has class constructors and class destructors, however.
My preferred way to implement singletons is to only expose an interface type that has the necessary properties and methods to store and retrieve the data, and a factory method that returns an instance of the interface. The class implementing the interface is private to the unit (declared and implemented in its implementation section). The factory method creates an instance of the class on the first call to it and stores the interface obtained from it in a variable only visible in the unit implementation section. The unit gets a finallization section that sets the interface to nil to finally destroy the singleton.