Java does not release oracle cursors

I am working on the following code below (edited for clarity) which gives me several problems with open cursors in Oracle. Basically I am trying to fetch data from DB and for each returned row, there are 0 or more rows of ancillary data that need to be fetched and appended to the record. This is currently achieved by calling another function when populating the main dataset to read auxiliary data. Which is great for small amounts of lines, less than 1000. While this is a normal operating range that users will probably use, they will probably ask for all lines, which could be on the order of 10 thousand lines. High volume execution fetches results in ORA-01000: maximum open cursors exceed error. If I run the code and query v $ open_cursors, you can see the cursor count untiluntil he falls.

If I comment out the line calling the subfunction, it works fine, the number of cursors in v $ open_cursors just hovers up and down by a few counts.

I noticed that the main function is to pass the connection object through a subfunction, and I thought that this could cause the resulting statements and target objects to remain open while the connection is still open, even if closed by code. So I tried to change the code so that each function would get its own connection from the pool and close it when it was done, but that didn't make any difference for cursors.

I could increase the number of cursors, but a) that just masks the problem, b) I have to stick to it to a stupidly large number to make it work, and c) I don't want to release flawed code!

However, I was running out of ideas on how to get the code to release cursors.

public ArrayList getCustomerSearchResult(Connection con) throws AnException {
    ResultSet rst = null;
    PreparedStatement stmt = null;
    ArrayList resultList = new ArrayList();

    String sql = "----  The search SQL string --- ";

    try {
        stmt = con.prepareStatement(sql);
        rst = stmt.executeQuery();

        while(rst.next()) {
            DataDTO data = new DataDTO();

            data.setSomeData(rst.getString("...."));

            // ##### This call is where the problem lies #####
            data.setSomeSubDataAsAnArrayList(getSubDataForThisRow(data.getId(), con));

            resultList.add(data);
        }

    } catch(Exception e) {
        throw new AnException("Error doing stuff", e);
    } finally{
        try{
          rst.close();
          stmt.close();
          rst = null;
          stmt = null;
        }catch(Exception ex){
            throw new AnException("Error doing stuff", ex);
        }
    }
    return resultList;
}

public ArrayList getSubDataForThisRow(String Id, Connection con) throws AnException {
    ResultSet rst = null;
    PreparedStatement stmt = null;
    ArrayList resultList = new ArrayList();

    String sql = "----  The search SQL string --- ";

    try {
        stmt = con.prepareStatement(sql);
        stmt.setString(1, Id);
        rst = stmt.executeQuery();

        while(rst.next()) {
            SubDataDTO data = new SubDataDTO();

            data.setSomeData(rst.getString("...."));

            resultList.add(data);
        }

    } catch(Exception e) {
        throw new AnException("Error!", e);
    } finally{
        try{
            rst.close();
            stmt.close();
            rst = null;
            stmt = null;
          }catch(Exception ex){
              throw new AnException("Error!", ex);
          }
      }

    return resultList;
}