try (Connection connection = dataSource.getConnection()) {
..................................
        } catch (SQLException e) {
            throw new IllegalStateException(e);
            // или просто прокидывать исключение наверх
        }

java.lang.Exception: Exception from method
	at console.Console03$Resource.doSomething(Console03.java:10)
	at console.Console03.main(Console03.java:21)
	Suppressed: java.lang.Exception: Exception from closeable
		at console.Console03$Resource.close(Console03.java:15)
		at console.Console03.main(Console03.java:22)

package console;

public class Console03 {

    public static class Resource implements AutoCloseable {

        public Resource() throws Exception {
        }

        public void doSomething() throws Exception {
            throw new Exception("Exception from method");
        }

        @Override
        public void close() throws Exception {
            throw new Exception("Exception from closeable");
        }
    }
    
    public static void main(String[] args) throws Exception {
        try(Resource r = new Resource()) {
            r.doSomething();
        } catch (Exception ex) {
            ex.printStackTrace();
        }        
    }
}