Why is my phone having issues charging?

@inspirinoxx17 you actually answered your own question. Your very last sentence "the piece where the cord plugs into the port is wobbly". Just because the cable does not fall out, does not mean that it makes proper connection with the little pins inside that connector. This also explains your phones not charging symptoms.

Luckily this is not a terribly complicated or even expensive repair. The charging board is soldered to the Sub board (daughter board), which you need to replace. Since iFixit does not yet have a guide for this, use the one from here It is written in Spanish but easily translates within most common browsers. A replacement board can be purchased from this great place ;-) Galaxy S20 FE 5G (US) Charging Port

While you work on your phone, take lots of pictures. Once you are done with it, take those pictures and create a guide for iFixit. That will help the next person that has the same problem as you do. It's easy and fun to create these guides. https://ifixit.com/Guide/new The iFixit community will appreciate this.

Repair is War on Entropy!