In such constructions, the usage of the infinitive is a bit special. A sentence such as “Je lui ai dit de venir” can only mean that the person you are talking to is doing the action. You are asking the person to come; you are not telling them that you will come.
Your first sentence should therefore read “Je lui ai dit que j'ai/avais encore besoin de cette fiche”.
In your second sentence you use an impersonal construction. As you noticed you have to use a full proposition with a conjugated verb: “Elle m'a dit qu'il me faut/fallait une nouvelle fiche avec la date d'aujourd'hui.” Dire is never followed by a verb in the subjunctive.
In both cases, the tenses of the subordinate clauses are often chosen to match with the one of the principal clause. The imparfait is the preferred form unless you want to particularly stress the fact that, as of speaking, the problem is still to be solved. In which cases you can use present or future tenses (the usage is quite similar to the English one here).