Hi, Gian. Glad to hear that your poly problem was solved.
On the next part of your problem, I would start by laying a 4 foot level on the gutters and be sure that they actually slope toward the downspout and don't have any dips in the run. Then take your hose and run some water in them to see if the weight of the water is causing them to sag. If this appears to be a problem, you can get a kit at the home center with long screws and aluminum ferrals to add more supports to the gutter. Be careful installing them from a ladder. The length of the screws makes this awkward.
After the water has run out of the downspout, try to push your hose down it with no water running. It should snake down easily. If not, try blasting the clog with the water running. At least, you should be able to tell where any clogs are located.
A lot of older houses has fresh water run-off lines that connect to the city sewers. The problem with these is that they were nothing more than clay drain tiles from the end of the downspout to the sewer and the tiles were just butted together. Obviously, this as a great invitation for any kind of roots to take up residence and clog the drain. If this has happened, the best choice is to replace the system with PVC piping glued together. The piping is usually not too deep so there is not a lot of digging, but do watch for other utility lines in the ground. Call "Miss Dig" if you have any question. The PVC pipe is pretty cheap, too. The "slope the ground away from your house" suggestion is OK in the suburbs where you have some lawn but it isn't practical where you have 6 or 8 feet or a concrete driveway between you and your neighbor.