In creating a language, varying sentence structure can be difficult, but is rather important if you want to make it truly different from your native language. There are a lot of small things in other languages that lend them distinction from our own.
Spanish possessives work this way. There are the possessive forms of "our" (nuestro) "your" (tu) and "his/her/its" (su), but very often they simply add "of he/him" (de él) after the noun being possessed. It's a simple change that gives Spanish a little more distinction from English.
Another option -and I'm not sure if it exists in real languages or not- is to use a sort of article/pronoun connection to define a possessive. In your language, saying "my hat" could translate literally to be "the me hat". "His horse" "The he/him horse" (depending on how to differentiate subject and object pronouns). "Your water" "the you water". You could also use actual possessive forms but still leave the article: "Her house" "the her house", "my mother" "the my mother".
What is interesting about possessives is that they have different forms of ownership. C.S. Lewis had this mentioned in The Screwtape Letters in that "my boots", "my dog", and "my God" are all possessive forms with different meanings. To add complexity to your language, you could create different words or sentence structures to identify this difference. From the previous language example, "my boots" would be "the me/my boots", but "my dog" might either stay "the me dog" or be changed to "he the me dog" to identify the relationship between dog and master. Boots do not change behavior based on who owns or wears them, but an animal, a slave, or a child are not possessed in the same way at an object. For "my God", we don't refer to our possession of Him, but His of us, so the phrase might turn as "me the He God" (an abnormal form, since everything previous has identified the owner, not the subject). To turn it around, "I am God's child" and "The Lord is my God" would read differently as "I am me the God(His) child" and "The Lord is me the He God". A third difference is one of equality with personality "my friend" is not my property, and I am not his, yet there is still a defined possession in the relationship. To acknowledge both parties involved, it could be spoken in a similar way to my dog as "we the me friend" allowing us to understand that there is more than one person involved.
I'm not sure if I'll use this in my own work yet or not. I'll let you know if the notion becomes free. Until then, I hope it helps to shed light on your options in linguistic engineering.
*brain fizzles and sparks* whoa. Awesome. 
Thanks for sharing!