"The lawn needs mowed" is grammatically incorrect in terms of the English language. I dunno if you learned another language first or if it's just a regional thing like how certain regions pronounce words completely different -- I saw your blog about it -- and if I'm honest I'm not sure what the general opinion is among English professors and the like about accents (where you pronounce a word differently in a non-native language because you're used to speaking the way you do in your native language) and dialects (where you just pronounce it differently because of your area or other influences, not because of another language). Since dictionaries define strict pronunciation rules and grammar is taught strictly within a language, I would assume the idea is that accents and dialects, per their nature, are technically "incorrect", but as with anything as long as the intent is understood then it's not a big deal. Grammar's usually a lot stricter than pronunciation though. People pronouncing things differently than others over long periods of time can basically make it the norm, and that's "correct", but that's not really immediate; I can't go and convince a bunch of people to pronounce "crouton" wrong and then insist that our new pronunciation is correct because it's not (I actually do pronounce crouton wrong on purpose for my own amusement though :xd: ).
I can honestly say I've never heard someone say "the lawn needs mowed" as common phrasing in an area, or anything like it, so I dunno what the case with that one is. If it's a common thing then I'd just chalk it up as basically some form of slang that most people just aren't aware of. Correcting it is arguably like correcting people saying "cool" then (before that was officially considered a word anyway), which would be silly, but if people don't even know about it as a term then people are going to correct it simply because they don't know it's common, or common in an area. I'd say just realize that people aren't gonna know that saying it like that is from your area.
As for how I say other things differently than some people... I don't really know because I'm not really aware of any dialects that might have influenced me while growing up. I generally strive to pronounce things "dictionary-correct" when I can, but there's a few cases where I pronounce things differently. I use more Canadian-English dialect pronunciations in a few cases, such as the word "aunt". Then there's the example of me mispronouncing "crouton" on purpose because I think it's funny.