I'm sorry, but I don't want (you) to use that tool.
つかう（use) + ない（not） = つかわない
つかわ is a conjugated form, which is used with ない.
Literally, it says "(I) want (you) not to use that tool".
Don't delve into thinking too much about the meaning of ほしい here
～ないでほしい is just a very very formal way of saying ～ないでください. It's Keigo. You can tell the sentence is already very formal from the use of もうしわけない, thus the use of ～ないでほしい is very natural here and keeping the formalities in all parts of the sentence consistent. もうしわけない doesn't go that well with てください
Another alternate and equally formal version of ～ないでほしい is ～ないでもらいたい
Keigo ftl
This sentence wants to say:
I'm sorry, but I don't want (you) to use that tool.
つかう（use) + ない（not） = つかわない つかわ is a conjugated form, which is used with ない.
Literally, it says "(I) want (you) not to use that tool".
Don't delve into thinking too much about the meaning of ほしい here
～ないでほしい is just a very very formal way of saying ～ないでください. It's Keigo. You can tell the sentence is already very formal from the use of もうしわけない, thus the use of ～ないでほしい is very natural here and keeping the formalities in all parts of the sentence consistent. もうしわけない doesn't go that well with てください
Another alternate and equally formal version of ～ないでほしい is ～ないでもらいたい