Can I Legally Break My Lease?

No, you can't move without breaking your lease and being liable for the rent.

You need to write to the management company of the property detailing the issues, and requesting a resolution. They then have a 'reasonable time' to resolve the issue.

You should NEVER sign a lease without looking at the property first.

There's one thing you can do. You can sue to have the lease voided by a judge. You can fully expect to lose. Judges do not void leases over very fixable issues. The Judge will order that the issue is fixed in a certain window of time, and if it isn't, you may be able to reopen the case.

You can get out of your lease but you will have to hire a lawyer to help you.  How you expect to do anything without money is beyond me.  Why did you strap yourself financially? Did you report back to the landlord that it needs a second fumigation?  That the bugs are still running wild?

Sorry but you should've inspected the place more carefully before you moved in. Since the landlord is making a reasonable effort to fix the problem, you aren't getting out of your lease for this. I suggest you spray more throughly and try to figure out how they are getting in and seal up the cracks in the walls, but that's really the best you can do. If the place were so horrendous that you called the code enforcement agent and he said the place wasn't up to code - they you could leave for this (assuming the landlord still isn't able to get the place up to code) but it doesn't sound like the problems are that bad.

You were a tenant from February 1. You don't rent from a realtor, unless it is the landlord's property manager. Your problem is bigger than security deposit. You are responsible for rent for the full lease period. Laws vary by jurisdiction (location). You send a written letter to the property manager with dead bug photos if possible asking to send a exterminator again. 

" I'm still finding dead ones inside the house in different areas."

Yes, that's what happens when you spray chemicals on bugs.  They DIE. 

And sometimes they die right where you can see them.

Of course, extermination only came ONCE.  You have only lived there two weeks!

STOP SPRAYING CHEMICALS ALL OVER.

Seeing bugs is WAAAAAAAAAAAAAAAAAAY safer for you and your young children then having an excessive amount of chemicals in the home!

Why not just go to Home Depot and ask them which of their products work best in your part of the country?  We got some white pills there that knocked them out for good, but they also have sprays, bombs, etc.  I would have done this before even mentioning it to the landlord. 

Of course not.................

No bugs will not automatically get you out of the lease without paying and NO you would not get your deposit back.  The only way out of the lease without paying is to sue them & get released by a judge unless the landlord agrees to it.  

You are legally required to give them an adequate chance to clear the infestation & 2 weeks does not cut it.  It can take up to 2 weeks just for the 1st treatment to work! You have not given them enough time to address the problem to legally break the lease without paying or get any money back.  

You first have to give your landlord a chance to fix the problems . You need to give him a written request .