If you are using a tiny pencil tip, there will be a significant initial drop in temperature when you touch the iron to the area to be soldered. The smaller the tip, the bigger the drop in temperature.
It is not that you should not touch the tip of the iron with solder, but ideally, you will have enough solder on the tip of your iron to form a thermal bond with the area you wish to solder. This will allow heat transfer into the joint, which you then apply the solder to. If the temperature is well matched to the melting point of your solder, and size of tip is large enough to supply enough heat to the body of metal you are trying to solder, the solder will melt and spread(flow).
There are several things that can cause you problems, dirt, insufficient heat transfer, mismatched iron/solder temperatures.
I suspect you are new to soldering, and you have asked a sensible question! You summarised that your solder is 700F = 371C so chose a setting on your soldering iron of 375C. That would seem absolutely logical in theory.
You have not said exactly which Elenco soldering station you have, or exactly which solder you have, as such there are many potential points to address here.
Using the Amazon iron as an example, of the 5 tips included with that iron, my feeling is that the 2.4D tip (Number 4) would be the most suitable choice, based on it is big enough to enable heat transfer but small enough to not be too large if soldering IC legs for example. I would have a better idea of which to recommend to you if I saw the tips in my hand, it may be that number 3 is suitable also, and would achieve a higher heat transfer. If number 3 is not too big for what you are trying to solder, try that one. Tips 1 and 2 are simply too small for general purpose electronic soldering, as heat loss is too high, and transfer too low.
Quality and type of solder varies tremendously and has a dramatic effect on the soldered joint. Everybody has gone 'lead free' mad, but lead free is not as easy for a beginner as traditional lead solder. I have had a soldering iron in my hand for about 40 years, and I am still using the same solder. It is the choice of professionals. Expensive, but ease and quality of finished joint speaks for itself. The solder I am referring to is Ersin multicore 60/40 activated rosin flux. Farnell element14 UK
The 60/40 505 No-Clean Flux or 60/40 511 No-Clean Flux types would be my recommendation.
Now we have identified a good quality suitable solder, let me draw your attention to some of the important points. Specifically for the 511. "High wetting ability - good spread on copper, brass and nickel - for fast soldering". You can just watch it spread almost instantly! The main point I wish to draw your attention to though, is the melting point of the solder, and the suggested iron temperature!! Then you will see why your initial guess at setting you iron was WAY OFF the mark...... For the 511 multicore solder the melting point is 180C with an iron tip temperature recommendation of 308C.
So I think now you can see, as is typical with these bundled irons and solders they are a good price and a good introduction to general purpose soldering, but some refinement will make your experience so much better. I could not see an actual specification for the solder Elenco supply, and you didn't specify which particular model of Elenco kit you had. So my recommendation would be get some good quality solder such as that I mentioned, (activated rosin cored 60/40 leaded solder) not necessarily ersin multicore brand. And get the largest possible tip size suitable for the style of soldering you are planning to work on.
Flux, can compensate for some dirtiness, but most important with soldering, is cleanliness of the joint!
What concerns me most about your statement of 700F solder, you run the risk of damaging IC's, transistors, Arduinos!, anything really if you are working at those sorts of temperatures. My main recommendation to you, get some suitable solder.
Regards,
Graham