need to create a validation rule for record type to restrict the user from creating leads with 2 conditions, one when the email is blank and another when (screening pick value is unscreened and Territory Lookup is blank), following is something I tried but it works for first condition but not the second.

AND(  $RecordType.Name = "Partner Record Type",  ISBLANK(Email), OR($RecordType.Name = "Partner Record Type",  ISPICKVAL(Screening__c, "Unscreened"), ISBLANK(Sales_Territory__c))))

need to create a validation rule for record type to restrict the user from creating leads with 2 conditions, one when the email is blank and another when (screening pick value is unscreened and Territory Lookup is blank), following is something I tried but it works for first condition but not the second.

AND( 
  $RecordType.Name = "Partner Record Type", 
  ISBLANK(Email),
  OR(
    $RecordType.Name = "Partner Record Type", 
    ISPICKVAL(Screening__c, "Unscreened"),
    ISBLANK(Sales_Territory__c)
  )
))