[AngelicaDeAlquezar]

"Fiarse" means to trust, like "confiar", but they use different constructions.
I wouldn't say that "fiarse" is an old fashioned verb; I see it commonly used, although maybe more often in higher register conversations or in written language. You may certainly use it. Yet, we don't really use it like in your example.

The usual formula is "fiarse de" + "algo" (not really "fiarse de que" + "acción").

- Eso me pasa por fiarme de su buena fe. = Eso me pasa por confiar en su buena fe.
- No te fíes de esa mujer; sólo te quiere por tu dinero. = No confíes en esa mujer...
- Los políticos se fían demasiado de las encuestas. = Los políticos confían demasiado en las encuestas.


However, with an action, the verb is quite commonly used in the negative form:

- No me fío de su palabra. = No confío en su palabra.
- No me fío de Juan; me parece un hipócrita. = No confío en Juan...
- No me fio de que sea honesto. = No confío en que sea honesto.
- No me fío de que el negocio prospere. = No confío en que el negocio prospere.
- No te fíes de que tus amigos te ayudarán. = No confíes en que tus amigos te ayudarán.


As for your examples: "Me fío de que no huya" sounds weird because, as I said, there is an action instead of "something" or "someone".

Your proposal, "confía en mí que no huya" is a slightly weird sentence. "Confía en mí" is asking the other person to trust in me, while "que no huya" seems to be expressing a wish that the third person will not run away, so there are two subjects that aren't related by any linking word.
If you say "confía en que no huya", this is an imperative sentence to ask my listener to trust the third person won't run away, so this doesn't express what you mean either.

We would rather say "Confío en que no huirá" or "confío en que no huya". (Personally, I prefer the first one, as the certainty of the verb "confiar" is reinforced by the indicative. I have nothing against the subjunctive though, which expresses the spontaneous shadow of a doubt.)


If you mean to say "I'll make sure they won't run away", you may say something like "me encargaré de que no huya", "me haré cargo de que no huya", "me aseguraré de que no huya", things like that.