Has anyone here ever replaced a thermostat?

[Paper Roses]

Something bizarre happened while I was away for 3 days. I keep the temp at 64 degrees in my house.
Came home to a 38 degree kitchen and an 42 degree LR.
The furnace did not kick on while I was gone. When I came home I goosed the dial to 65 degrees and the furnace started right up and has seemed to keep going as always.
Don't know if this was a funny glitch or if I have a problem

It is $125.00 to have the oil company come to check things out.(This does not count the cost of parts or repair) Can't afford it. Can find no reason this should happen. Any ideas?

If it happens again, could it be the thermostat and is it a do it yourself project? I have a 6 year old dial type unit. When the furnace was replaced (6 years ago), they put in a digital unit. It turned on and off every few minutes. Too sensitive, I had them put in one of the older round type and I'd like to keep this dial type if that is the problem.
So far no more bizarre incidents.
Sound like anything familiar?

Could this old lady replace it herself? I have no idea what it involves.