Okay, let’s see if I can help.
I’m pretty sure こく getting a dakuten in words doesn’t follow a pattern and is, in fact - random. (If it isn’t then I’m making a pretty big fool of myself right now.) So you should try and learn the words that use ごく to not mix anything up again!
The words containing 国 (こく or ごく) you should have seen by now are:
米国 (べいこく), America
中国 (ちゅうごく), China
外国 (がいこく), foreign country
天国 (てんごく), kingdom of heaven
全国 (ぜんこく), nationwide
As you see, you have two words with rendaku, heaven and China. Everything else uses こく.
I remember it like that: Heaven is pretty special. It’s nothing that can be found on earth. It gets rendaku’ed for its specialness And China, well, China is China. I don’t have any trick to remember this, it just happens after repeating the word often enough.
To overcome that problem, I wouldn’t focus on seeing a word as “two kanji”. I would try to look at it and identify it as “one word” so that you don’t have to think about the kanji readings for too long and end up putting the wrong one.
It’s a pain, but you have to learn when to use 日 as じつ or にち. I can help you out with 月 though. Every specific month (January, February, March…) uses がつ. The question “what month?” (何月) uses がつ as well.
Words like “last month” (先月) or “this month” (今月) use げつ.
Did you use the mnemonics? I thought they weren’t helpful at all for memorizing those words so I just made a list (one thing, two things…) and studied them everyday over a short period of time and then it clicked :)) ひとつ、ふたつ、みっつ…
Sorry for the long reply… there’s not that much to read actually