It does not hurt and may well help a lot to be completing all your practice tests closed book and under timed conditions. However, for sure by July you must be writing closed book. June studying may still be effective, even if you need to do some brief consultation of books and notes while writing.
In June, most people are still learning the law. (Yes, we call it “bar review” but there is a great deal of new information that most of us learn while studying for the bar exam.) One great way to learn rules is to type them out (or better still, handwrite them 50 times). So, if you are writing a practice exam, and know there is a rule but forget how to phrase it, or do not know what the rule is at all, go ahead and look it up. Then, close the books, and try to type it into your answer from memory. After the brief book consultation to refresh your recollection (pun intended), finish writing your analysis and conclude on the particular issue you are addressing.
Stopping to look up rules will not likely be too much of a crutch so long as you give the full month of July to entirely closed book writing. Having some open books in June will still help you learn the law while you practice exam writing. And, you are likely to more easily recall rules in the future that you learned in the context of particular fact patterns.
When July hits, close the books while writing practice exams. If you don’t know a rule, try hard to remember whatever you can of it and/or analogize to an area of law you do know and write what you think the rule likely is. The minute you finish the essay (writing under timed conditions), then look up the rule or rules you guessed on. Right then and there, as you are studying a model or sample answer, (which you must do to get the full benefit of practice writing), write out any rules you missed. Write them on your flashcards or outline, say them out loud, and/or re-write them 10-20 times to “seal” them into your memory. (Everyone memorizes differently.)
By allowing yourself to look up some rules in June but targeting July for being closed book, you are giving yourself a “break” of sorts while maintaining a concrete goal and acknowledging the need to ramp up as the exam approaches.
The bar exam is at the end of July, not tomorrow. You still have plenty of time to hone your analytical and writing skills, to vastly increase your knowledge of the law, and to memorize those rules you will need to articulate in bar exam essay writing. Keep at it! Slow and steady wins this race.