One of the initial external links shared in the replies had helpful info but personally I prefer for standalone answers/snippets to be right here on the Stack Overflow question page. See below snippet for what I used and solved my (similar) problem. If anyone has problems (or adjustment suggestions) please chime in.
IF EXISTS (SELECT * FROM sys.objects WHERE object_id = OBJECT_ID(N'[MyLockedAndDelayedStoredProcedure]') AND type in (N'P', N'PC'))
DROP PROCEDURE [GetSessionParticipantAnswersFromEmailAddressAndSessionName]
GO
CREATE PROCEDURE [MyLockedAndDelayedStoredProcedure]
@param1 nvarchar(max) = ''
AS
BEGIN
DECLARE @LockedTransactionReturnCode INT
PRINT 'MyLockedAndDelayedStoredProcedure CALLED at ' + CONVERT(VARCHAR(12),GETDATE(),114);
BEGIN TRANSACTION
EXEC @LockedTransactionReturnCode =sp_getapplock @Resource='MyLockedAndDelayedStoredProcedure_LOCK', @LockMode='Exclusive', @LockOwner='Transaction', @LockTimeout = 10000
PRINT 'MyLockedAndDelayedStoredProcedure STARTED at ' + CONVERT(VARCHAR(12),GETDATE(),114);
-- Do your Stored Procedure Stuff here
Select @param1;
-- If you don't want/need a delay remove this line
WAITFOR DELAY '00:00:3'; -- 3 second delay
PRINT 'MyLockedAndDelayedStoredProcedure ENDED at ' + CONVERT(VARCHAR(12),GETDATE(),114);
COMMIT
END
-- https://gist.github.com/cemerson/366358cafc60bc1676f8345fe3626a3f