Brother, this has been made to be confusing quickly when it's not plus some incorrect things have been said. I can't go over all the posts and respond, but here's kinda the story.
You should NOT upgrade a branch circuit receptacle to 20 amps unless the supply wire feeding it (inside trailer) is rated 20 amps and the breaker is 20 amps. What if someone came along and plugged in an appliance with a 20 amp rating and the wiring in the walls was only 15 amps (#14 ga.)?
You should NOT have to install a 20 amp receptacle if you think the contact pressure inside is not enough. If you really think that's issue (and I highly doubt it) you could install a 15 amp commercial spec. (aka "spec. grade) receptacle. These have a higher contact pressure. Or if you think that's still not good enough, you could use a "hospital grade" recept. which has even higher contact pressure. The hospital grade ones are really hard on the hand to push/pull a plug....
If you find the plug and/or recept. are warm, take a look at the plug. The prongs may need cleaning. Or more likely, especially if it has been used outdoors, the conductor stands are corroded. This is quite common. In that case, you need to cut back the cord until you find shiny copper and re-attach the plug.
If the campground voltage does indeed drop, this does NOT cause the amperage of the heater to increase. Portable heaters are purely resistive. Some heaters are rated dual voltage, such as in 208 volt for commercial use and 240 for residential. You will find a sticker on them 208/240 volts. Heater output varies as the square of the voltage. Here is a dual rated heater from HD: http://tinyurl.com/B-B-heater
208/240 x 208/240 = 0.75. So in the case of a 2000 watt heater rated at 240 volts, it becomes a 1500 watt heater if connected at 208 volts. This is basic Ohm's law. So a 2000 watt heater at 240 volt draws 8.3 amps while connected at 208 volts draws 7.2 amps.
I really do not think that RV type receptacles are a problem in any way. The problem is when the wires are NOT terminated properly on the push-on "stabs." I pulled two receptacles out of the wall in our TT and found that on one of them, the conductor on one of them was not properly terminated and only one of the two stabs was connected. These RV receptacles are made by reputable wiring device companies like Pass & Seymour and meet all required safety standards and codes. It is commonplace nowadays for receptacles in houses to have "push in" wiring connections on the back instead of, or as well as, screw terminals. There's no way I will use those push-in connections. The contact pressure simply can't be as high as two stab connections on the back of an RV recept. Yet they meet all required standards.
Receptacles do not last forever and with a lot of use, they can wear out. Especially the el-cheapo 69 cent HD ones. I always go to an electrical wholesaler for electrical supplies and get commercial grade or premium residential grade items.
I am pretty certain that the OP has a cheapie heater with a 16 gauge cord which will definitely get warm or hot with continuous use. Assuming it has an appropriate safety sticker, I'd at least consider replacing the cord on it with a 14 gauge. Still not good for a heater to be running continuous though so that's only part of the answer.
BTW, feeding a 30 amp RV panel with a 15 amp rated extension cord is not a problem whatsoever as long as the breaker at the supply end of the extension cord is 15 amps. You *could* have a 1000 amp breaker in your RV if you wanted, fed by a 15 amp extension cord as long as there is that 15 amp breaker at the begining of it all.