I don't know- I guess I'm just not so freaked out about it that I would panic along with everybody else here.
Yes you do have a problem, and yes it needs to be addressed ASAP or sooner, but.....
First of all it is a fifty year old Remington. The trigger,as you said, is very light. No box stock, over the counter Remington has ever been designed with a trigger that had "light" design parameters. If it isn't at least two pounds or so, take it to a gunsmith and have him adjust it for you, and check the sear engagement at the same time. Too light a pull and/or too little sear engagement can cause any Remington (or other rifle ) to fire on bolt closing.
If you think the bolt is oily/greasy/dirty......it is. Dis-assemble it and clean it thoroughly. If you don't know how, have someone who knows show you how, or you can have it done fairly cheaply. Re-lubricate it with a good synthetic before re-assembly. Keep in mind that how your bolt internals look is similar to what your trigger may look like inside. If your bolt has fifty year old grease in it, so does your trigger. Have the smith check it/clean it before he adjusts it. If you don't trust the trigger, get a good aftermarket. If you have to have it light, make sure the adjustment range on the trigger will handle it.
If it all is just too much and you want to sell the gun, that is up to you.