Question: In Ex. 34:1 God says he will write on the stone tablets, but in Ex. 34:27 he tells Moses to do the writing.
Response: The word "write" is being used in two different ways. In the first verse, it is being used in the sense of being the author, or originator, of the 10 Commandments. In the second verse, the word "write" is being used in a physical sense, in which Moses is inscribing the words of God into the two stone tablets.
God is the author of the 10 Commandments, Moses was the inscriber. God is writing the words through Moses.
Here are the NIV English translations of the two verses in question:
The LORD said to Moses, "Chisel out two stone tablets like the first ones, and I will write on them the words that were on the first tablets, which you broke. - Exodus 34:1 (NIV English translation).
Then the LORD said to Moses, "Write down these words, for in accordance with these words I have made a covenant with you and with Israel." - Exodus 34:27 (NIV).
The first verse shows that God is the author of the 10 Commandments, and the second verse shows that he worked through Moses to inscribe the words of the 10 Commandments.