We will have to replace the old tv soon, when the prices go down.
This could well be your best option. A new converter box will run you about $50, and if you are going to replace that TV soon anyway, you will just be throwing away that $50. Your old TV could even die before you get a new TV. Fortunately, this is a great time of year to be looking for a new TV. Many stores have last year's models on clearance at really good prices. Check some stores. My 2 local Walmart stores each had at least 20 of various sizes on clearance yesterday.
Until then, what can I do to fix this?
Try, in this order:
First, converter boxes have an "energy saving" feature that shuts them off after a period of time. Go into the menu and make sure that this feature is disabled, or set for the longest time period available.
Second, you can try unplugging the unit for an hour and see if that "resets" it.
Third, the unit may be overheating. Does it get very hot when it is on? if this is the case and 1 and 2 above did not help, you may have the box in a confined area without ventilation. Try moving it to a better spot. If there are any openings in the case for ventilation, make sure they are not obstructed or dusty. As a last resort, you might try ventilating the case by putting openings in the plastic case. Be careful when doing this to avoid damage to the internal components.