(concept question) What is the correct way to add digital signature with RSA?

What is the correct way to add a digital signature in a pdf? in practice. Do we save the private key in our company's database because it can be used to decrypt or verify a signature later. since only the public key can be used to encrypt data. please offer some advice.

const fs = require("fs");
const path = require("path");

// display pdf buffer
const pdfBuffer = fs.readFileSync(path.join(__dirname, "test.pdf"));

// generate a rsa key pair
const { generateKeyPairSync } = require("crypto");
const { publicKey, privateKey } = generateKeyPairSync("rsa", {
  modulusLength: 4096,
  publicKeyEncoding: {
    type: "spki",
    format: "pem",
  },
  privateKeyEncoding: {
    type: "pkcs8",
    format: "pem",
    cipher: "aes-256-cbc",
    passphrase: "top secret",
  },
});
const text = "Name of the person signing the document";

// encrypt text
const { publicEncrypt } = require("crypto");
const encryptedText = publicEncrypt(
  {
    key: publicKey,
  },
  Buffer.from(text)
);

// decrypt encrypted text using private key
const { privateDecrypt } = require("crypto");
const decryptedText = privateDecrypt(
  {
    key: privateKey,
    passphrase: "top secret",
  },
  encryptedText
);

//  add the encryptedText to the pdf buffer
const pdfBufferWithEncryptedText = Buffer.concat([pdfBuffer, encryptedText]);

//  save the pdf buffer with encrypted text
fs.writeFileSync(
  path.join(__dirname, "test-encrypted.pdf"),
  pdfBufferWithEncryptedText
);