Can the past tense of an i-adjective and the past tense of a verb be in the same sentence? For example: "I bought a book that was heavy."
Does it sound unnatural?
It's not prohibited, but it never ever means what past tense in English does. Tense in Japanese subordinate clause is (basically) relative to main clause, so if you bought a book and the book was heavy, you just have to say 重い本を買いました.
重かった本を買いました suggests the book was heavy before you bought it. But there rarely are books that being sometimes heavy, sometimes light, like King Arthur's sword. A more likely setting is that you bought that book on the premise that it was heavy, for example:
You went into a bookstore. You took books in your hand. This book was light. This one was light too. Oh, this one was heavy. Thus, you bought it.
This sentence is unnatural. "I bought a book that was heavy" is translated as "重い本を買いました".
Sequence of tenses isn't necessary in Japanese. For example, "I read an interesting book yesterday" is translated as "私は昨日、面白い本を読みました".