Your realtor will simply present you a qualified file (good credit, proof of income, landlord references, etc.) and you'll say yes or no and then it's just sign the lease and give them keys.

Am I cutting it too close?

Regarding the remaining partner's partnership, putting very simply. 

The previous landlord, simply wasn't one.

If they're firm on the price, this could result in an upside down first deal for you with numbers that tight - simply not enough room for "uh-oh's" or "we better fix that" scenarios. 

My wife and I have been landlords for 16 years and now have three properties simply because we just converted our homes to rentals each time we moved. 

You also have to consider that the additional monthly payment will cut into your cash flow.

Go to the website for the town> then the Tax Assessor page> then look for a link to online property information databaseThere are several different companies providing this service in MA, so it is easiest to simply start with the town website and go from there. 

Are there hard money lenders which simply loan to the LLC entity?

Simply waiting 30 days from purchase is not sufficient.