question

Well, if you were on the right track, then you would be using SoC in your program.

https://en.wikipedia.org/wiki/Separation_of_concerns

None of the database code should be in the button_click event, the event should have called a method that had the database code and let the database method do the work on the behalf of the button_click event.

Now with the database code being in its own area, the database code can be called from anywhere within the program by any form in the program and even from a Form_Closing event.

This may help you, and I say go back to the starting blocks and get a little basic understanding. You seem to be sharp minded and you should be able to easily pick it up.

https://www.codeproject.com/Articles/36847/Three-Layer-Architecture-in-C-NET-2

HTH

If you want what was retrieved next time the form is opened create a setting under project properties. In the following simple example there is a setting LastCustomerIdentifer with a default value of -1 which indicates it has not been used.

Notes

• All code presented requires C#8 which will work with C# 9 too
  

• Only reason for a singleton is you mentioned using one, there really is no reason to use a singleton as static methods in a class will work just fine.

Singleton class

 using System;
 using System.Data.SqlClient;
    
 namespace SimpleConnectSqlServer.Classes
 {
     public sealed class DataOperations
     {
         private static string _connectionString =
             "Data Source=.\\sqlexpress;Initial " +
             "Catalog=NorthWind2020;Integrated Security=True";
            
         private static readonly Lazy<DataOperations> Lazy = new(() => new DataOperations());
         public static DataOperations Instance => Lazy.Value;
    
         public string CompanyName(int identifier)
         {
             using var cn = new SqlConnection { ConnectionString = _connectionString };
             using var cmd = new SqlCommand
             {
                 Connection = cn, 
                 CommandText = "SELECT CompanyName FROM Customers WHERE CustomerIdentifier = @Id"
             };
                
             cmd.Parameters.AddWithValue("@Id", identifier);
             cn.Open();
                
             return (string) cmd.ExecuteScalar();
         }
     }
 }

Form code

 namespace SimpleConnectSqlServer
 {
     public partial class Form1 : Form
     {
         private int _iterations = 6;
         public Form1()
         {
             InitializeComponent();
             Closing += OnClosing;
             Shown += OnShown;
         }
    
         private void OnShown(object? sender, EventArgs e)
         {
             if (Properties.Settings.Default.LastCustomerIdentifier <= -1) return;
                
             CompanyNameLabel.Text = DataOperations.Instance.CompanyName(Properties.Settings.Default.LastCustomerIdentifier);
             IdentifierTextBox.Text = Properties.Settings.Default.LastCustomerIdentifier.ToString();
         }
    
         private void OnClosing(object sender, CancelEventArgs e)
         {
             if (Properties.Settings.Default.LastCustomerIdentifier > -1)
             {
                 Properties.Settings.Default.Save();
             }
         }
    
         private void GetCustomerNameButton_Click(object sender, EventArgs e)
         {
             if (!int.TryParse(IdentifierTextBox.Text, out var id)) return;
                
             CompanyNameLabel.Text = DataOperations.Instance.CompanyName(id);
             Properties.Settings.Default.LastCustomerIdentifier = id;
         }
     }
 }

Verify in SSMS

(SQL-Server Management Studio)

Settings tab under project properties