The 3.4 vs. 3.8 issues is hammered out here:

https://camaroforums.com/forum/93-02-v6-tech-14/3-4-vs-3-8-a-55344/

 

When your searching for a 4th gen for your mom, she would be fine with either the 3.4 or the 3.8 they are both good engines. Also be sure to check out the rear frame rails on the cars you look at cuz i was looking at a few that looked nice but when i checked the rear frame rails, i could actually fit my fist into the rust holes in the frame rails.

 

If you look around for a while, you can find many deals. I payed 3200 For my 3.8 With 281 Miles. Yes 281 Not 281,000. The Previous Owner had GM replace it and his daughter hated Putting the carseat in it so they just wanted it gone. I had gotten it from ebay, But there are deals everywhere if you look hard enough! Best of luck for your searches and For the $500 You get