mamatobe92 I read your post. I hope you have success soon with a new rental. I am not surprised you are stressed. Please try not to worry. Yes you will be restricted on what you can find as some landlords just don't want pets in their property and of course with a dog and a baby access to outside space would be really beneficial.
Some things that you could consider which might help speed up your search. You may have already done all this or know all of this so I don't want to come across as condescending but just in case, this is what I have done in the past/ would do:
Do make sure you are registered with all estate agents locally to you. It would help if you or your partner could make an appointment and see then in person and explain your deadlines if you haven't already done this. From these appointments you can ensure they have your deadlines. You can also get more info on the estate agents process for applications - they'll all be slightly different. They may only process rental apps on a specific day a week so do check this for each estate agent.
You need to find a sympathetic person in one of the estate agents who will be on your side and help focus your search and be proactive for you. You can then keep in touch with them twice a week on the phone. Make sure you phone them twice a week. Phone them first thing when they open. Follow up on all applications & give feedback on all viewings.
If you're missing out on rentals you are applying for, are you getting feedback from the estate agent so you know if there is anything that you can change on your applications for next time? Is it timing of the application that is the issue? Can you apply online to speed things up (make it easier for you)?
If you're on facebook then ask on your local town sites whether anyone has a ground floor flat or small house available for rent coming up in the next few weeks/ months. This way of looking can be surprisingly quick and effective. Give them your budget and timescales. You don't have to share where you are currently living of course. Don't be afraid of repeating your request...still looking locally.. Perhaps find a picture of a rental you have missed out on and really liked and say you're looking for this sort of property then you have a picture to add to facebook which will make your post more noticeable. This way of searching for property can be really good - and quick and a lot less stressful.
Don't give up!
Hopefully your partner can do the running around and most of this for you.
Good luck with your pregnancy.
Primrose