I want to use it to mean "thoroughly make use of ... without pushing it/them over its/their limit" but I am worried it does not mean that.
Yes, exploit can carry a negative connotation of using someone else for your advantage and even being horrible to them.
Instead of exploit, I would say:
I benefited from x's help
This sounds much nicer than I exploited x and now does not carry a negative connotation but instead, it shows you made use of their help.
If you want to emphasise how much help they have you, you could say:
I thoroughly benefited from x's help
This implies they benefited and helped you a lot, and you did not push them over their limit.