Check with your local codes .. I have a feeling whoever told you that you aren't allowed to dig in the utility easement is wrong. I could be wrong though due to your codes. where I live you must call in for utility locates but digging is okay. As long as they do there locates they aren't liable for any damages that you may cause. I've even dug across the states natural gas pipe line 10' in diameter only at 3' in depth ( with of course the gas guys watching me ). As a irrigator- while digging I run into utilities more then you can imagine.
Well for one-I'm a irrigation expert and not drainage expert. But like most things in life- to solve the problem is how much you are willing to invest into it.
A simple effort would be to install a drainage system around the perimeter of the fence 2' down with perferated pipe 1" drain rock / sand and tie into your existing drain. A company could easily do that in 3/4 day-full day .
A advanced effort would involve replacing all of your soil in the back with a sandy loam and 4" of top soil ( Actually not as big of deal as it seems ) with an elaborate drainage system to collect and divert water run off.
Unfortunately - there is no simple solution that will just 'fix the issue'