Sometimes a component fails because another component is deteriorating. You replaced R618 but the root fault apparently is elsewhere.
Did the previous R618 appear burned? If so then it could have failed because of overcurrent / overvoltage coming from a component which is gradually becoming shorted internally. Or there's a device which is supposed to draw away current but it has gone open. Etc.
You report seeing the current rising. This suggests the problem gets worse as temperature rises. Maybe there's a transistor near R618, whose bias has increased gradually until the transistor goes into thermal runaway?
Examine components in the area. Look for burn marks, smoke marks, cracked solder joints. Etc.
Capacitors are known to get leaky (particularly electrolytics). Do you see a bulge on a cap anywhere?
Does your scope use tubes? A bad one may start conducting too much current.
If you feel like sacrificing more fuses, try putting your finger on various components near R618. See if one gets too hot to touch.
Do you have a freeze-spray? It might help you locate the bad component by keeping it cool so the scope continues operating.
It may be tempting to install a fuse rated much higher than spec, so the scope will continue operating even with the faulty device, or until the device fails. This is risky because it can ruin other components.