The present habitual ...mar bím tuirseach should be used here. Tá mé tuirseach is a simple statement of fact at the present moment, but codlaím is a habitual action. Imagine codlaím was qualified temporally and the sentence was, Codlaím gach oíche mar tá mé tuirseach. This sentence would clearly be nonsense because it's translation is "I sleep every night because I'm tired (right now)." Bím tuirseach fits in much better because it's meaning is roughly "I'm tired (every night)." This remains true even if there's no explicit temporal adverb, because of the habitual aspect inherent in codlaím and bím but absent from tá mé
To clarify what I mean by present habitual is that it refers a discrete series of actions, some which have already occurred and others which haven’t but are likely to occur in the future. This contrasts with a (true) present that refers to some state of affairs currently ongoing without any reference to past of future.
As a verb codlaím can only be habitual, because for most verbs the Irish present tense is really a present habitual tense i.e. the present habitual interpretation is the only possible one. If you wanted to specify that the sleep is currently happening you’d have to say Tá mé i mo chodladh. This is true for most verbs in English too, “I sleep” can only be interpreted as a general statement of behavior, as opposed to “I’m sleeping” which describes what I’m doing right now.
Where English and Irish differ, though, is the verb “to be”. In English “I am” admits both a (true) present and a present habitual interpretation depending on the context e.g. true present in “I am so tired I could go to bed right now” and a present habitual in “I am tired everyday after work”. In Irish though the verb bí has different forms to capture these two different meanings. The true present uses táim as in Táim chomh tuirseach sin go rachainn a chodladh anois which corresponds to my first example, while the present habitual form of bí is captured by bím as in bím tuirseach tar éis obair an lae.
What's key to understand, and what I think isn't sufficiently well explained, is that it's the táim form that's exceptional, not the bím form. Every verb has present habitual forms, no other verb has special forms for the true present.
Tá tuirseach orm is even worse, as tuirseach is in fact an adjective, and the Tá ... ar construction requires a noun - Tá tuirse orm.
But while other adjectives like brónach, ocrach, etc are normally only used when translating attributive adjectives from Irish (Tá an fear ocrach ... - "the hungry man is ...", "hungry" is an attributive adjective) relying on a noun construction when translating a predicative adjective (tá ocras ar an fear - "the man is hungry" - "hungry" is a predicative adjective), there are a number of examples of tuirseach being used as a predicative adjective in the FGB, so your LC Irish teacher might have been a bit reactionary.