Using VLOOKUP [ or INDEX (MATCH) ] to Find Two Separate Occurrences in an Array

I am struggling to properly define VLOOKUP [or INDEX (MATCH)] to find two separate occurrences of the same event in an array. Here is the background:
1) I have three tabs in my Excel File entitled “BY DATE”, “YOUTH”, and “ADULTS”.
2) Two (2) Adults and two (2) Youth participate in an event each week and I record the date they participated next to their name in the appropriate tabs for “ADULTS” and “YOUTH”.
3) The “BY DATE” tab is designed to populate the names of which Adults and which Youths participated on a given week, listing the dates chronologically down the first column (reference my simplified Excel Forum Example file).
4) I am trying to use VLOOKUP in the “BY DATE” tab to search the “YOUTH” and “ADULT” tabs for a given date and place the names in the appropriate locations for a summary view.
5) Let me explain with a verbal example: On May 15, 2017, YOUTHs Hayley and Michael participated with ADULTS Ben and Debbie in the event. I inserted the date by their names under “YOUTH” tab and “ADULTS” tab, respectively (Highlighted GREEN).
6) In the “BY DATE” tab, I search the “YOUTH” and “ADULT” tabs for the date May 15, 2017 to find the names:
Formula: [ =VLOOKUP($A4,YOUTH!$A$3:$B$12,2,0) ] AND [=VLOOKUP($A4,ADULTS!$A$3:$B$12,2,0) ]
7) MY PROBLEM: I have found that each time VLOOKUP searches the array it starts from the beginning of the array ($A$3) and consequently finds the “first” date occurrence each time.
8) I believe I can solve this if, for the second YOUTH and second ADULT, I could make VLOOKUP begin the array search from the line after the first YOUTH or ADULT found. For example, on the “YOUTH” tab the VLOOKUP array search [$A$3:$B$12] identified Hayley on row 4 for May 15, 2017. To find the second YOUTH, I need the VLOOKUP array search to start with the row after Hayley; i.e., row 5 ( the VLOOKUP array should look like [$A$5:$B$12]). I do not know how to automatically increment VLOOKUP to begin the second search on the row following where the first name was identified.
Can anyone help and recommend a solution to my problem? THANKS IN ADVANCE FOR YOUR ATTENTION.

FlameRetired: I want to thank you for responding to my post so quickly. I must admit that I am fairly novice in Excel. You have provided a formula using functions (and "array-entered formulas") that I am not familiar with. However, please don't misinterpret - I am grateful that you would help and suggest this approach. I enjoy learning new things in Excel. Allow me to study the formula and the operations of the functions to see if I can grasp the concept. Hope you don't mind if I perhaps come back and ask few questions