I have not tried Jackson. I just have similar situation.

I used org.apache.commons.text.StringEscapeUtils.unescapeJson but it's not working for malformed JSON format like {\"name\": \"john\"}

So, I used this class. Perfectly working fine.

https://gist.githubusercontent.com/jjfiv/2ac5c081e088779f49aa/raw/8bda15d27c73047621a94359492a5a9433f497b2/JSONUtil.java

// BSD License (http://lemurproject.org/galago-license)
package org.lemurproject.galago.utility.json;

public class JSONUtil {
  public static String escape(String input) {
    StringBuilder output = new StringBuilder();

    for(int i=0; i<input.length(); i++) {
      char ch = input.charAt(i);
      int chx = (int) ch;

      // let's not put any nulls in our strings
      assert(chx != 0);

      if(ch == '\n') {
        output.append("\\n");
      } else if(ch == '\t') {
        output.append("\\t");
      } else if(ch == '\r') {
        output.append("\\r");
      } else if(ch == '\\') {
        output.append("\\\\");
      } else if(ch == '"') {
        output.append("\\\"");
      } else if(ch == '\b') {
        output.append("\\b");
      } else if(ch == '\f') {
        output.append("\\f");
      } else if(chx >= 0x10000) {
        assert false : "Java stores as u16, so it should never give us a character that's bigger than 2 bytes. It literally can't.";
      } else if(chx > 127) {
        output.append(String.format("\\u%04x", chx));
      } else {
        output.append(ch);
      }
    }

    return output.toString();
  }

  public static String unescape(String input) {
    StringBuilder builder = new StringBuilder();

    int i = 0;
    while (i < input.length()) {
      char delimiter = input.charAt(i); i++; // consume letter or backslash

      if(delimiter == '\\' && i < input.length()) {

        // consume first after backslash
        char ch = input.charAt(i); i++;

        if(ch == '\\' || ch == '/' || ch == '"' || ch == '\'') {
          builder.append(ch);
        }
        else if(ch == 'n') builder.append('\n');
        else if(ch == 'r') builder.append('\r');
        else if(ch == 't') builder.append('\t');
        else if(ch == 'b') builder.append('\b');
        else if(ch == 'f') builder.append('\f');
        else if(ch == 'u') {

          StringBuilder hex = new StringBuilder();

          // expect 4 digits
          if (i+4 > input.length()) {
            throw new RuntimeException("Not enough unicode digits! ");
          }
          for (char x : input.substring(i, i + 4).toCharArray()) {
            if(!Character.isLetterOrDigit(x)) {
              throw new RuntimeException("Bad character in unicode escape.");
            }
            hex.append(Character.toLowerCase(x));
          }
          i+=4; // consume those four digits.

          int code = Integer.parseInt(hex.toString(), 16);
          builder.append((char) code);
        } else {
          throw new RuntimeException("Illegal escape sequence: \\"+ch);
        }
      } else { // it's not a backslash, or it's the last character.
        builder.append(delimiter);
      }
    }

    return builder.toString();
  }
}

Extract From Jettison:

 public static String quote(String string) {
         if (string == null || string.length() == 0) {
             return "\"\"";
         }

         char         c = 0;
         int          i;
         int          len = string.length();
         StringBuilder sb = new StringBuilder(len + 4);
         String       t;

         sb.append('"');
         for (i = 0; i < len; i += 1) {
             c = string.charAt(i);
             switch (c) {
             case '\\':
             case '"':
                 sb.append('\\');
                 sb.append(c);
                 break;
             case '/':
 //                if (b == '<') {
                     sb.append('\\');
 //                }
                 sb.append(c);
                 break;
             case '\b':
                 sb.append("\\b");
                 break;
             case '\t':
                 sb.append("\\t");
                 break;
             case '\n':
                 sb.append("\\n");
                 break;
             case '\f':
                 sb.append("\\f");
                 break;
             case '\r':
                sb.append("\\r");
                break;
             default:
                 if (c < ' ') {
                     t = "000" + Integer.toHexString(c);
                     sb.append("\\u" + t.substring(t.length() - 4));
                 } else {
                     sb.append(c);
                 }
             }
         }
         sb.append('"');
         return sb.toString();
     }