I know it’s wrong, but I do it all the time or else my sentences would go on forever.
What makes you think this is an error? All the greatest writers of English have started sentences with and. Mark Liberman, linguistics professor at the University of Pennsylvania wrote about this mythical “rule” in Language Log in 2005:
There is nothing in the grammar of the English language to support a prescription against starting a sentence with and or but—nothing in the norms of speaking and nothing in the usage of the best writers over the entire history of the literary language. Like all languages, English is full of mechanisms to promote coherence by linking a sentence with its discourse context, and on any sensible evaluation, this is a Good Thing. Whoever invented the rule against sentence-intitial and and but, with its a preposterous justification in terms of an alleged defect in sentential “completeness”, must have had a tin ear and a dull mind.
So, my answer to OP’s original question is “mu”: your question assumes something which is false.
The rule forbidding conjunctions at the beginning of sentences—“No Initial Coordinators” (NIC)—is something that even the most prickly old usage writers have rejected as Not A Real Rule. Arnold Zwicky wrote on Language Log in 2006 (emphasis added):
Mark notes that the AHD [American Heritage Dictionary] note for and rejects NIC out of hand, and he provides a smorgasbord of cites (and statistics) from reputable authors. Similarly MWDEU [Merriam-Webster Dictionary of English Usage]. Paul Brians, collector of common errors in English, labels sentence-initial coordinators a "non-error". Bryan Garner denies, all over the place, that NIC has any validity. Even the curmudgeonly Robert Hartwell Fiske tells his readers that there’s absolutely nothing wrong with sentence-initial coordinators. A point of usage and style on which Liberman and I and the AHD and the MWDEU stand together with Brians and Garner and Fiske (and dozens of other advice writers) is, truly, not a disputed point. NIC is crap (emphasis added).
But still it lives on, as what I’ve called a zombie rule. It’s been lurking in the grammatical shadows for some time — at least a hundred years, to judge from MWDEU. Hardly any usage manual subscribes to it, but it is, apparently, widely taught in schools, at least in the U.S., with the result that educated people tend to be nagged by a feeling that there is something bad about sentence-initial and (and but and so).
A participant using the name pointytilly links to Paul Brians’ list of "non-errors" in defense of the view that sentence-initial and and but are "grammatically correct". And indeed they are, according to essentially everyone with any plausible claim to expertise, prescriptivists and descriptivists alike.