here is nothing to indicate the tense in this sentence. I wrote my name wrong is grammatically incorrect. Wrong qualifies how I wrote my name and is therefore an adverb. I write my name wrongly or I wrote my name wrongly should be acceptable. If you don't like wrongly you could use, 'incorrectly'.
DanielCark is right. Wrong is grammatically correct here. Wrong is like 'fast' or 'straight' and is a reduced adverb of manner that does not take an -ly ending. It's one of a handful of exceptions to the general rule.
You wouldn't say "We drove fastly" or "They went straightly to school" or "Tom never does anything rightly."
Here is what the Oxford English Dictionary has to say about these adverbs under 'adverbs of manner'. (Also applicable to American grammar)
If we say "saya menulis nama saya salah" it may be the exact translation , but it sounds weird and we dont say it indonesia, we put "Salah" after "saya" , then it be "saya salah...".
We rarely put adverb in the end of sentence , except we add by "dengan" eg: "saya menulis nama saya dengan benar" = I write my name correctly .
But we never say "dengan salah" cause generally "wrong/salah" is accidental
Now you're wrongly accusing the adverb of being archaic. Wrong and wrongly are both used in American and British English.
However wrong is also used as an adverb: