Turn your main breaker off in your house, check meter to see if you are still "using" power.
Did that and it still appears to be using power, blinking one square of four on the meter.
Yes, the faster the dots move from left to right the more load/usage, could be meter specific. If you are showing usage with the main turned off you have an issue.
There is a few reasons that can be checked by a quality electrician. I have seen in the past, are usually related to corrosion. On the lugs, in meter base and main panel, if using aluminum wires, there should/needs to be anti-oxidant. If none was used or little was used after time corrosion builds up. When your meter was replaced this may have unintentionally jarred the connections caused this to surface. Underground wiring of service entry into the house not in conduit. After time and cold cycles will cause some issues but since this happened after a meter change my money would be in the meter base. You most likely will be able to see the corrosion on a lug.
If you could leave your main off for an hour two, write the numbers down on meter and then turn off main, write numbers down just before turning on to see usage during that time. It may not show too much because there wont be much load. As load/usage increases the corrosion/resistance will increase causing more usage. The longer you could leave off, the more accurate you will be.
This is just my internet opinion and may be way off base since I cant see it or test myself .