This is more of a style question than a linguistics question. Both sentences are ok, of course-- English speakers wrote them and they generally seem to convey the ideas intended. One may be better in a particular context or based on a particular convention, but one isn't "better English" than the other. As for whether you may have been taught (or otherwise found) rules for how to link sentences, it's likely they were guidelines rather than real rules, to help struggling writers, more than to tell you something you can't do. A good rule with that is to be aware of conventions, suggestions, "rules", etc., and then only deviate from them if you have a good reason. If so, go for it. Don't start a sentence with a conjunction (and, but, or...) unless you want to. But if you want to, go ahead. You probably have a reason for doing it. You might be points taken off if someone conservative is grading your essay, so be aware of any conventions you should be following (and who your audience is).
As for this specifically, I can't really comment on it from a linguistics perspective. This is at the level of discourse and implies a relationship between the two sentences (within the domain of Pragmatics); it does not state such a relationship explicitly, not in either example sentence.
What you are describing might be related to the idea of "tail-head linkage" found frequently in discourse structures for some languages where one sentence is related to the next explicitly by repeating or referring to some part of the previous sentence. Something like "I arrive. After I arrived, I did something else." -- not very elegant in English, and varies by language, but the idea is that it is all explicitly connected. That isn't quite what's going on here but could be interesting for you at least.
