IF MOISTURE in in the tank is the problem, you need to do whatever it takes to resolve it.
First, two lines out of the top of the tank, indicates this is the type of pump setup with a return line. THAT IS GOOD because it is easier to prime the pump (mostly "self priming")
GENERALLY these type tube connections use what amounts to a compression fitting such as you would use in an icemaker water connection, except that the fitting has been "drilled through" so that you can simply drop the line to the bottom of the tank right through the fitting, and then tighten it up.
I WOULD INVESTIGATE whether the line(s) are TOO CLOSE to the bottom of the tank. Do what you need to pull the lines out. You probably will have to cut the lines about a foot or more out of the tank, and buy two compression unions (probably 3/8?) to reconnect the tubes.
Use something to measure the tank depth, and bend the tubes so they are 2-3" off the bottom OR MORE.
(READ ON but for now, if you can NOT clean the tank as per below, pull these lines up from the bottom a foot or more until you can clean the tank)
Examine the fittings on the tank--the filler, the gauge, and the line connections. If the tank is under the eaves of a house, especially, examine the tank fittings to see if there is any way that rain/ snow can get into the tank
NEXT make sure there is a method to drain the VERY BOTTOM of the tank. The tank should be pitched slightly towards whatever is there for a plug/ valve/ drain. If there is no valve and cap there now, so you can drain the "sump," PLAN to drain the tank, say, in the summer, and install one.
For now, if there is no drain, remove the fill plug/ cap, and get some hose/ tubing and some container so that you can insert a tube clear to the bottom, and siphon some out. A good way to do this is to buy some 3/8 or 1/2 tube so that you have a rigid probe to get to the bottom of the tank, and get some of that cheap clear plastic hose so that you can see the water/ dirt/ oil and the flow. Get yourself one of those siphon starter pumps, at any auto parts store.
Now simply siphon until it's clear.
To purge the fuel system, after you have siphoned the tank bottom, disconnect your new union at the return line at the tank--you'll have to follow this line from the filter to determine which one. The filter will be in the SUCTION line, but you want the RETURN line
So disconnect your new union at the return, at the tank, and have someone down at the switch at the furnace. Run the return into a bucket until it's clear, then change the filter and nozzle what you hope to be ONE last time.
Depending on what kind of pump you have, there PROBABLY is a filter IN THE PUMP. This needs to be serviced, preferably by someone who knows how.
Oil furnace nozzles are incredibly small, and you MUST have ABSOLUTELY clean oil