They sound like SLA / AGM (lead acid) batteries. The rest of this post is going to assume that's what they are (so if they're not, just let us know).
They don't always provide much range even when actually new, and "new" ones often sit on a shelf uncharged for so long that they are as damaged as old ones people are replacing, so "new" ones often don't fix problems like yours. Some have a manufacture date stamped into the plastic, many don't--if yours do it can be a guide to how old it is, and potentially how long it sat around unused wasting away.
You can try buying from a different seller, to see if you get lucky and get fresher ones that will last a little longer, but a different battery type will probably be a better solution, and have a longer lifespan (and possibly better range than you had with the originals).
Another problem is that at high discharge rates (like those your scooter draws), they don't supply anywhere near their 'rated" capacity. The graph below shows this, and some other things, including that at 12v resting they're getting closer to empty than full. (12v under load isn't bad, but your 12v is with no load; you're just seeing them bounce back to 12v after the load is removed--you probably would see not much more than 10v at best under load when the scooter stops working). It's worse at lower temperatures (graph for that at the link).
http://www.polluxbattery.com.my/battery-discharging.html
THe "C" numbers (3c, 2c, etc) mean how many amps the load (scooter) is pulling from the battery, compared to how many Ah (Capacity) the battery has. So at 1C that's 12A for yours (1 x 12Ah). At 3C that's 36A.
Your scooter is "1000w", which if that is how much power it's using during the trip (might not be that much; depends on speed, riding conditions and terrain, etc) would be around 28A or so (1000w / 36v = 28A), so at that rate, by that chart, with brand new perfectly working batteries (that haven't sat on a warehouse shelf for who knows how long), you might get 10 minutes of operation; less as the load goes up or temperature drops, or the batteries age, etc. Since the scooter probably doesn't pull that much current all the time (usually only at startup from a stop and then while climbing hills or going up slopes, etc), it should last longer than that with new batteries, probably two or three times that long.
Other battery types don't have that problem, are lighter for the same capacity, and let you use more of that capacity (as long as you don't get a badly made one, whcih unfortunately are common in the cheap stuff).
So...depending on the layout of the batteries in the scooter, you may have enough room for a good Lithium battery; you'd also be replacing the charger (because it uses a different kind than the SLA).
You'd need one that can deliver at least 30A continuous, and that has at least 10Ah (partly for range, partly so the cells aren't being pushed that hard). The more Ah, the more range you can get out of it, and the easier it is on the cells inside (they sag less in voltage under load, so it performs better and lasts longer).
It will be called a 10s 36v battery for the voltage you need. It may also be called 10sXp (like 10s3p), where Xp means how many parallel cells it has in each group of cells inside--the actual number doesn't matter, since it is relative to what cells the pack is made of and what it's specified to be able to deliver in A and Ah.
If you're interested in doing this, and can post the dimensions of the inside of the battery compartment (good pics help too), along with your budget, we can try to help you find a suitable one.
There are also some SLA-sized Lithium replacements; some of them may be suitable for your scooter and you'd just use three of them just like you use three SLA now.
It's possible there is some other problem, but if it appears to work the same way it did originally, except it doesnt' ahve as much range, then the battery is almost certainly the cause of that (it's the only thing that makes sense, unless you also have some other problem not reported in the first post). So if there is other info not already provided, or your battery is not SLA, let us know so we can help with that.