Hi Vee,
I hate to tell you, but the myth that rice will absorb moisture from the phone has been thoroughly busted; in addition to not helping, it can introduce contamination into the parts of the phone you're trying to clean so it may actually make things worse. A better choice would be a bag full if silica gel packets, which will absorb moisture with none of the disadvantages of rice.
But just as effective would be to simply use a hair dryer to heat up the port and dry it out.
Your instincts regarding cleaning are spot on, though. The recommended cleaning liquid is 90% or higher concentration of isopropyl (rubbing) alcohol; wet an eyeglass cleaning cloth with it then wrap it around a toothpick and use that to wipe the contacts and clean out any lint or dirt inside the port. Examine it carefully with a magnifying glass and a bright light to check for any corrosion that's built up on the pins inside the charging port and make sure there aren't any bent or broken pins.
If none of that works, then it's most likely time to replace the port. Unfortunately it doesn't appear that iFixit has gotten around to creating a guide for that repair, but there's a Spanish site called Nadie Me Llama Gallina that has just what you need.
[link|https://www.nadiemellamagallina.com/en/manuales/smartphones-1/apple-5/apple-iphone-13-pro-max-1324/flex-de-carga-11942|Manuals / Apple iPhone 13 Pro Max / Charging Flex - Nobody Calls Me Chicken]
The site is in Spanish, but used with the Chrome browser's translation it is quite usable no matter what your native language, and their guides are well written.
With regard to buying the part, if you want to support iFixit's repair efforts you can buy it right here.
[product|IF262-067|iPhone 13 Pro Max Lightning Connector Assembly]
Their parts are of the highest quality and they're warrantied for life, but they can admittedly be a bit pricey, so if cost is of concern, you can also find plenty of other sources for less money on sites like Amazon, eBay and AliExpress, among others. Note that it comes in various colors to match your phone, but realistically the only difference is the inside of the charging port, so in practical terms any color will work.
As you work, take lots of pictures and when you're done you can use them to create your own iFixit guide to that repair for those folks who come after you with the same problem. It's easy, fun and will earn the respect of the repair community.
[link|https://www.ifixit.com/Guide/new|Create a new Guide - iFixit]