Seems to me it's no surprise that an oil company wants to get a lease on it's terms, which are rarely in your interest as a mineral owner. I've seen enough one-sided leases to be wary of them, and your attorney surely didn't suggest the modifications for nothing.
In addition, you have to consider the fact that a lease is "forever", i.e. if oil or gas are found and produced - in some leases not even in paying quantities - the lease can be "held by production" for an unlimited time and bind not only you but your descendents.
That's why I'd personally rather forgo a lease on bad terms. If there's enough interest in the area, there will always be more chances.
Naturally, if you are in need of cash at this particular time, that can override any long-term thoughts. But it's always worthwhile considering that a landman is trying to get the best deal for his client, and that is not you, but the oil company. You, on the other hand, are trying to get the best deal for yourself .
The landman's comment that Chesapake would probably "sit on" the lease, resell it and you would get "much less" sounds like pure bluff. In the first place, they wouldn't buy a lease they didn't like any more than you would lease at terms you don't like. And if they did, you've got the bonus already and the terms of the lease are binding on any subsequent owner of the lease as well as Chesapake. So it's a mystery to me how you could lose anything that way.
As for the landman not returning calls or emails... I can't say that's anything new to me. My advice would be to write him that if you don't hear from him within a certain period, you consider the signed lease invalid and will feel free to deal with other possible lessees. If even that gets no action, wait for other offers. In that case, the landman has probably gone on to other owners who are willing to sign on the dotted line without even reading the lease, and has no further interest in you.