Hello and welcome to JustAnswer. Please note:This is general information and is not legal advice. No specific course of action is proposed herein, and no attorney-client relationship or privilege is formed by speaking to an expert on this site. By continuing, you confirm that you understand and agree to these terms.
I am sorry to hear about this situation. If the lease is not signed, then it is by default a month to month lease. If so, then you can get out, but only after giving them a month's notice.
Also, understand that unless it was in writing signed by both parties, the restriction on pets arguably does not apply. However, the landlord is likely to press the issue anyhow and seeming as you are unhappy, it may be most practical to simply give 30 days notice and then terminate the lease with the standing that this is a month to month lease.
If the landlord attempts to keep the deposit claiming that this was not a month to month lease or another reason, one can go to Court to seek the deposit back and let the Judge decide.
I hope this helps and clarifies. Please use the SEND or REPLY button to keep chatting, or please RATE when finished. You may always ask follow ups at no charge after rating. Kindly rate my answer as one of TOP THREE FACES/STARS and then SUBMIT, as this is how I get credit for my time with you. Rating my answer the bottom two faces/stars or failing to submit the rating does not give me credit and reflects poorly on me, even if my answer is correct. I work very hard to formulate an informative and honest answer for you; please reciprocate my good faith with a positive rating.