I've deleted all messages and cleared the web history, and there are only about 4 photos on there. It tells me to delete apps but when I try to, they're all the ones that came with the phone and I can't.
Has anyone else had this problem and what can I do about it? Other than restore it to factory settings and lose all my contacts?

restoring factory settings should not lose your contacts if you are signed in to google.

what storage exactly is full?

The internal memory when I go online with it, also the text messages are full if there are more than two on there. And I mean my phone numbers, not Google contacts. I can write them down then try it though.

i know you mean phone numbers. on android phones they are sync'd to your google account. if you have it turned on.
go here:
https://www.google.co.uk/contacts/

and if you can see all your contacts you're grand. they will be restored after a reset when you sign back in to google.

text messages are tiny in the scheme of things. sounds like you have something saved that is taking up a lot of space. but without seeing it don't know that i can help you.

Go into settings and then Storage and it should tell you what space is used up by what. Take a screenshot and post it here if you don't understand what it all means.

Which model is it?

It's not very specific, it just says how much space is available.