@Christian Wendelboe to answer one of your original questions, yes you can switch lenders mid-transaction, but you and the new lender have to get on your horse to get things done by your closing date. The biggest thing is the appraisal, you need to make sure the new lender orders that ASAP. If you're going to do that, you need to have your Realtor reach out to the other side to see if they're ok with delaying closing potentially. If they're not because it was a competitive situation and they have other offers to fall back on, you may be stuck.
Otherwise, I'm in the Local Lender boat (for Single Family), in that if you have a lender from the area you're looking in, listing agents and owner-occupant sellers are more willing to "trust" your offer. A lot of people have been burned by the "Out of State Buyer", who comes in with a high offer with financing from an out of town lender nobody has heard of, only to have them ask for tons of money off after inspection and/or the lender isn't able to close for some reason. If you have a local lender and a local Realtor of course, the listing agent knows who they are, Title knows who they are, etc. With the slowdown, this may not apply as much anymore, but it was definitely a thing the past couple years. If you're looking for Multifamily/Commercial (or the Single Family is already a rental), then @James Wise is correct in that they don't care about where your financing is from, so it may be worth the effort to look nationally for the best terms that apply to you and your investing.
I will also vouch for First Ohio, many of my clients use them up here in Toledo and they do work all over Ohio. Sorry your offer didn't get accepted, but there's tons of properties out there. Good luck!