In Apex, is it possible to use Polymorphism and Overloading to mitigate the need for conditional logic?

Yes, I would say there is a much better way. Use the Id.getSObjectType method.

public String getAddress(Id recordId)
{
    if (recordId == null) return null;

    SObjectType idType = recordId.getSObjectType();
    // further logic
}

Now, if you can set up your helper classes to just use an empty constructor, you could dynamically instantiate them based on this information. Just have them implement a common interface.

Helpers

public interface AddressHelper
{
    String getAddress(Id recordId);
}
public class AccountHelper implements AddressHelper
{
    public String getAddress(Id recordId) { /*implementation*/ }
}
public class ContactHelper implements AddressHelper
{
    public String getAddress(Id recordId) { /*implementation*/ }
}

Dynamic Instantiation

static Map<SObjectType, Type> sObjectToHelper = new Map<SObjectType, Type>
{
    Account.sObjectType, AccountHelper.class,
    Contact.sObjectType, ContactHelper.class
};

public String getAddress(Id recordId)
{
    if (recordId == null) return null;

    SObjectType idType = recordId.getSObjectType();
    if (!sObjectTypeToHelper.containsKey(idType)) return null;

    Type helperType = sObjectTypeToHelper.get(idType);
    AddressHelper helper = (AddressHelper)helperType.newInstance();
    return helper.getAddress(recordId);
}

The above is a bit more verbose than strictly necessary, but I believe it will be more instructive as such.

Hope this example will help you

With condition:

public class exec{
    public void execute(String procType){
        if(procType='PROC1'){
            //do this 
        }

        if(procType='PROC2'){
            //do that
        }
    }
}

With polymorphism:

public class exec{
    public void execute(IProcess proc){
        proc.execute();
    }
}

public interface IProcess{
    void execute();
}

public class Process1 implements IProcess{
    public void execute(){
        // do this
    }
}

public class Process2 implements IProcess{
    public void execute(){
        // do that
    }
}