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If the amount they're paying for closing costs doesn't matter too much to you, then stick with the lender you're comfortable with. It may not hurt to do an application and see if the rate they quote is possibly better than your current lender. I am in the same process as you, signed a contract for a new construction home. I had shopped the lender myself first and was also recommended to use the builder's lender. I am sticking with my original lender as I just feel more comfortable. To answer your question, no there aren't any other benefits other than their offer to pay closing costs. This is simply a deal the builder and the lender workout for the builder to try and direct more business their way.
That is what I thought. I was afraid of all the credit inquiries but from what they are saying, it doesnt impact my scores if I do within 30 days and its a mortgage. Im watching my scores like a hawk because that is the delicate situation in my journey. I was happy to see that my scores went up 35 points after my closed discovery card was paid. I paid it like 2 weeks ago in direction of my loan officer. I had no idea that it was affecting my score that much!!
Yes you can apply for both without an affect on your scores. You will see separate inquiries but as far as FICO scoring it will be treated as 1 inquiry