It sounds better to say that you will 'do' your best, instead of saying that you will 'try' you best. This is because, as you have first explained that you are tired, to say that you will try could be interpreted to mean that you may be too tired to try properly. However, if you say you will 'do' your best, it explains that despite being tired, you are going to complete the job, or whatever it is you have promised to do.
「頑張ります」は I’ll do my best = 「最善を尽くします」が一番近いかなと思います。
I’m sleepy ＝ 眠い
ちなみに、微妙に状況が違って、
「よく眠れなかった（から眠い）けど」は I didn’t sleep well, but 〜
「（眠いので）もうちょっと寝た方がいいだろうけど」は I need some more sleep, but 〜です。
To go for something means to try determinedly to do something.
"John is really going for the 100 metre sprint this year. I think he'll win!"
Regardless of = without thinking about
「go for something」は、固い決意でそれ（something）に臨むことを表します。
"John is really going for the 100 metre sprint this year. I think he'll win!"
（ジョンは今年100メートル短距離走にすごい決意で臨んでいる。勝つと思うよ）
Regardless of = 考えないで
I'm quite sleepy but will try my best nonetheless.
I'll give it my utmost despite my sleepiness.
There are many expressions in English that express doing one's best. When trying to convey you are tired but will still try your best you may refer to any of the following phrases;
give it my all- An expression to explain that you will do your best.
try my best- to give one hundred percent.
give it my utmost-To give your very best.
英語で「ベストを尽くす」という意味の表現はたくさんあります。
疲れているけどベストを尽くす、と伝える場合、以下のフレーズが使えると思います。
give it my all （ベストを尽くす）
try my best （100％のベストを尽くす）
give it my utmost （最上級のベストを尽くす）
Well you are tired then you can also say you feel 'sleepy' or just 'feeling tired'
If you are telling someone you will do your best you can say you will 'try your best' or 'do your best'
眠いことは、'sleepy' や 'feeling tired' で表せます。
「最善を尽くします」は 'I will try my best' または 'I will do my best' と言えます。
I'm having trouble keeping my eyes open, but I'll do my best.
I'm tired, but I'll do my best.
The two sentences you see provided above are excellent ways to express to your listener that you are sleepy but you will do your best. In the first sentence you will see the term trouble keeping one's eyes open. This means to be tired or sleepy. This term is appropriate for informal settings, like talking with close friends or family. It would make a great addition to your vocabulary.
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