Reset your user profile. See Frequently asked questions - General - The Document Foundation Wiki - important: rename the folder, don’t delete it. Resetting the user profile solves many problems.
You have to assign a language to your text, or the spell checker won’t work. You can do that in tools - Options - Language settings - Languages - Default Languages for Documents. But you cal also modify the Default paragraph style, select the Font tab and select a language there.
Please note that the spell checker doesn’t automatically correct any errors, it just highlights them. Lowercase changing to uppercase is Autocorrect’s work. Take a look at Tools - Autocorrect - Autocorrect options to see what it will do for you (and disable anything that you don’t like).
EDIT: Make sure that hunspell is also ticked, that’s the actual spell checking module. There’s a bug in Apache OpenOffice, and maybe in Libre as well, that messes up spell checking for files created in Windows Explorer (the file manager, not the Internet browser). Can you get the spell checking to work in files created within Writer? If you can, upload a sample file for testing purposes.
You can also try Tutorial Spell check and Language configuration - and check in at that forum to get more help. When you register, you need to provide a (fake) version of OpenOffice that you use, a trick to defeat spam bots.