Made a mistake in purchase contract, how do I correct it?

[Dhwani Shah]

Hey guys. So my wife and I are in the process of buying a home for ourselves. We have one investment property, but this is the first home for ourselves. The listing price for this was 320k, but we ended up getting it for 362k with buyer knowledge inspection and 25k appraisal gap. With the market being this crazy, we knew something this aggressive is only what's going to get us into the running. According to our agent, there were multiple offers and everything, but they ended up taking ours as it was the best. One of the other things we were okay with and was in the contract is a possession clause. So we have set the closing date to first week of June and we're going to take possession first week of July. Now this is where we messed up...we were under the impression that even after closing, the owners would pay for everything until we take possession. I just found out his wasn't the case talking with family and my loan officer. So as the contract is written, we're giving these guys essentially free stay (they only pay utilities) until they move out, which is June 25, so end of month essentially. This was not explained to us by our agent and we were okay with this because we thought they'd be paying for everything until we take possession. Currently we've done inspections and all came clean and the bank just converted the loan over to this property this week. What should we do now? Talking to the agent, he's like this is what got you guys this deal and going back would kill the deal and etc? I want to believe my agent, but we actually have talked to the owner as we ran into them during inspection and realized that some of the stuff they felt (eg. such as things they want to leave behind for us) was not the same as what we got back from our agent when we asked them. And conversing with the owners we think was a much better experience. 

I don't mind losing this deal (I still love the house, so don't want to, but if it's a must) and losing the inspection cost we incurred, but I don't want to lose this really good interest rate we had locked! Along with that, once the closing is done, what about the liability and how the house is taken care of and everything else. It's essentially ours, but as much as I want to trust the owners, I don't know them. They can break something, leave things behind, could get in an accident, and all that. Along with that, we're paying for everything after closing which I don't like either. What should we do? I'm thinking we can ask for rent or payment or we move the closing date to when they're moving? Is this also normal to give people possession for so long after closing? Just so much liability issues I see as well. And finally, I really want to just talk to the owners rather than going through my agent as I think they're good people.

Sorry for the long post guys, but really really appreciate any help and feedback on what's normal and what should we do. Thanks!