As you have described this problem it is most likely the igniter (the thing that glows). That is what provides the resistance to provide the current for the coils to open in your gas valve. When they break down they will still glow but the heat lowers their resistance and therefore the current they provide the coils. If it is bad a visual inspection will show that a great deal of it is dull. It should have "sparkles" throughout the entirety of it. If it is not dull your problem is in the 2 coils for the gas valve. Many places will try to sell you the entire valve. Don't buy that. You only need the coils/solenoids. This is a $25 or less repair if you do the work. This era of Kenmore was manufactured by Whirlpool. [http://fixitnow.com/2003/11/appliance-tip-of-day-dryer-disassembly.htm|Here is the gist] of the tear down procedure (It has been the same from 1959 - ?). If you find it is not the igniter or coils the problem will be in the timer. Last I remember they were about $100, but that has been a few years ago.
'''TOP 3 Likely Causes'''
1) Bad igniter
2) Weak gas valve coils
3)Timer