Punctuation between independent clauses

I want to use use correct punctuation, but sometimes I think the rules are too rigid. The rules say that you can (1) use periods after each independent clause (2) comma and coordinating conjunctions between independent clauses (3) semi-colon between closely related independent clauses. (4) semicolon plus conjunction. Fine, BUT, sometimes it seems to me that any of those methods are too severe. They seem too strong and are too intrusive. They rudely break the flow. Sometimes it seems to me that closely related independent clauses are best separated by commas. I admit that many authorities say that commas may be used for short independent clauses, but sometimes even long clauses are so closely related that commas seem to me to be the most natural and graceful punctuation to use. Now, whatever anyone says, I'll probably be back because I now feel that it will probably be necessary to supply examples of what I mean. Unfortunately, I don't have any at hand, but now that I know of this site, I'll be collecting them for future examination. Oh, I just noticed that this forum is about British English (Okay, nobody's perfect). I'm still interested in your opinion.