Love telling/writing stories except I am the absolute worst at both spelling along with punctuation. Normally I care less about these things because I can still read them although many people get extremely angry and dislike that I do this. I currently use the free version of Grammarly for it to fix my writing/texting and I feel embarrassed about it. Is there a way to fix these naturally or is it something all writers go through every day fixing their mistakes?
I don't think you should feel ashamed of using Grammarly to correct mistakes. You should, however, take it with a grain of salt. Not because it makes a lot of mistakes, but because you wish to learn correct spelling, punctuation and grammar it suggests. So, instead of blindly clicking on the red line try to fix the mistakes on your own before going for its solutions.
Secondly, there is a lot of advice to be found by googling your question. I didn't try any of it so instead, I will recommend a book by William Strunk Jr. called The Elements of Style. It focuses on creative writing and covers punctuation, using active voice and formating dialogue.
The last piece of advice I would share is to read as much as you can. Not only to read but to take it slow. Go through the sentences and ask yourself why did the writer do certain things. Why did he put a comma there, how did he format dialogue, etc?
The purpose of writing is not for you to be perfect, it is for your prose to be (close to) perfect. No writer ever spits out perfect prose on the first go. It just isn't possible. That is why the writing gods created the act of revising to move the imperfect closer to perfect.
Grammarly is a useful tool in this process. So are dictionaries. So are beta readers. So are spreadsheets, and checklists, and a dozen other things. Each of these, properly used, will improve the quality of the writing.
Rejoice in the savory sausages that come out of the process rather than the less savory input to the process.
Good grammar was created to help us communicate our ideas accurately and effectively. In general, using good grammar will improve your writing.
In general, though, the perfect grammar community is just another religion designed to create a club from which they can exclude others to make themselves feel included. (Other such clubs are etiquette, Ivy League schools, etc.)
Always be aware that sometimes the rules will interfere with communicating your ideas. ("This is the type of bloody nonsense up with which I will not put.") In those cases, don't worry about it. There will always be someone who will point out flaws in your work.
Writing is about heart. Use Grammarly if you wish, but anyone can follow the rules. Create your prose and hire a proofreader to fix it later.
Don't feel bad. Few people, if any, are perfect writers. As an editor, I'd rather fix spelling and grammar and other technical aspects of writing for someone who can tell a story well than struggle with a poor storyteller who is technically a good writer.
For example, I'm a competent writer with a good handle on PUGS (punctuation, usage, grammar, and spelling) but I dearly wish storytelling came to me as readily as writing mechanics do. The reality is, you write the best stuff you can and then work with editors and proofreaders to help make it the best it can be.
Partial answer about punctuation.
The most "natural" way to improve would be to read a lot of books that interest you (as in, books that definitely went through editing) until you get intuitive understanding of the most commonly used punctuation points. This really is a long process though, and if you want to learn faster you need to combine it with reading about punctuation and applying what you learned to your own text.
For example, if you are concerned about using commas, read about commas and try to find places in your text where commas should be. Don't try to remember everything in one go, just start with the rules that concern you the most and don't be discouraged if you have to look up a rule more than once.
If you want to learn how to do something, learn from those who know how to do it well, and practice doing it well. This used to be called "copywriting", and it's a good way to really internalize the language you're trying to express.
Mine the vast expanses of literary treasure for building blocks to create your own masterpiece.
I have read somewhere, "There are no writers, but only re-writers." While mapping one's thoughts in writing, or pre-writing, one doesn't need to be perfect in spelling, grammar, etc. Yes, while reviewing it, if one can refine it in terms of grammar, spelling, punctuation etc., without taking someone's help, it will be great. At the same time, together with one's creativity, or knowledge of a domain, it one also has reasonably strong grammar, that is great. Dependence on any software makes 'DEPENDENT' which may be avoided, if possible.