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Blocked storm water drain

I have 2 storm water drains that have become blocked overtime with debris. These are earthenware drain pipes. I have just moved into the property over 4mths ago and noticed that before the wet weather occurrence over these past 2 weeks, these pipes were dried with debris. I have tried on my own to dig out as much debris as i could before requesting the service of a plumber in the hopes of unblocking the drainage. After it flooding dreadfully, the plumbing service I called, were unsuccessful of unblocking either of the drains explaining that it is possible the drains lead to nowhere as they were just digging through dirt. They suggested that i should dig a large hole and insert a barrel into the earth to collect the storm water that way to spread across the under the earth. I feel that i might need a 2nd opinion. I dont want to pay a high sum of money i dont have for a suggestion like this if there could be other solutions or perhaps better options that could be more cost effective. I am unsure as to how long these pipes may have been left in this condition for these blockages to have occurred but I am hoping there might be a more efficient way. The plumbers suggested they would contact the council to request a storm water drain report to know if there are any drains that lead from these pipes as they are adamant there is not and digging a large hole would be the best way to resolute this issue. Should I call a plumber or a drainlayer or both? Any possible help would be appreciated.