Mr./Mrs. Customer, this isCustomer
A simpler method of determining if your cans are IC Rated, meaning they are suitable for direct contact with insulation is to remove the lamp (and possibly the trim) then look at the can housing. If it is silver (unpainted aluminum) it is IC Rated and there is no need to take a trek to the attic.
If on the other hand the can is painted white, it is not IC Rated and insulation within 3 inches of the sides or on top of the can could be part of the problem and a fire hazard as well.
Many times though, this is caused with a lamp that is too large (in watts) for the fixture. Each fixture has a label inside of it that shows the maximum size of lamp for a particular trim.
If the lamp size is higher than the fixture is rated for the thermal protector will do what you have described. When the fixture gets too hot the thermal protector opens turning off the lamp then after cooling down the protector closes again turning the lamp back on. If let go long enough this cycle can damage the thermal protector or increase the risk of fire.
Either way, there should not be any need to change the trim. Many lamp manufacturers have LED lamps available in the style that looks like your existing incandescent lamps and will work with your existing trims.
Please let me know if you have any other questions.