Not lumpy*, grainy. As in there are tiny bits that just won’t dissolve. Is this the cheddar, or is there some way to fix this?
*no pun intended. Or wanted.
Use Velveeta. It really does melt into a smoother sauce.
As you discovered, real cheese is hard to get completely smooth.
Needs dairy. Or Velveeta. 
It’s a cheddar thing.
Another vote for Velveeta. Sauces are where it really shines, and not because the fat has separated from the curds as happens with Cheddar.
It might help if you favor us with a list of your ingredients, as well as your technique. Please also include information on how the ingredients are stored before you begin cooking.
One of the first threads I started was on the subject of my cheddar cheese sauce being excessively grainy. It came down to the fact that the bechamel sauce I used for the base was itself grainy. I stopped freezing the butter that I brought into the house, and began taking my time with cooking the roux. This fixed both the bechamel and the cheese sauce.
Of course, if your cheese sauce is not based on bechamel, the fix will probably be different.
For most cooks, who are limited by time, using Cheddar is their downfall. A bit of Velveeta for texture, and a lot of Bleu or other flavorful, but assertive, cheese to offset the amateurs, should work.
Okay, I’m a Cooking Idiot, demonstrated many times. There are cheese sauces NOT based on a bechamel sauce?
Are you using pre-shredded cheddar? Some of those use anti-caking additives that can give the cheese a gritty texture.
I was using the most basic recipe possible: block cheddar cheese grated by hand, cream and butter. I heat the cream and butter in the microwave, and add grated cheddar a little at a time, beating thoroughly with a wisk, rewarming as necessary to keep the cheese melting. No oil separation at all (I wisked the heck out of it), it’s as smooth and homogenous as can be- EXCEPT for the tiny bits of grit.
Had the block cheddar spent any time in the freezer?
Sounds like a fluke to me unless you used a block of cheese that had been opened and you foolishly grated some of the hardened cheese. By the way, I like sour cream rather than cream if doing a quick cheese sauce. Worth a try if you like sour cream.
I counteract this problem by making a sort of cheese gravy. Start with a roux ( melt butter or margarine in the pan, then add just enough flour to make it pasty), stir in 1/4 cup milk, then turn the heat to low and stir in the cheese.
I believe that’s the roots of a bachamel sauce. Roux plus milk = bachamel sauce.
Add cheese and it becomes a Mornay sauce.
I’ve had trouble with frozen cheddar cheese sauces. I’ve found that you need a stiff, strong bachamel sauce to start with as your foundation… Mixing in some american cheese (I tend to buy mixed cheese ends - the bits at the end of the loaf that are too small to slice any more - at the deli, for all my cheese sauces, including that used for Mac & cheese. Mac & cheese sauce gets some onions & minced garlic sauteed into the bachamel and flavored with some mustard powder.)
How aged was the cheddar? The longer its aged the more moisture it loses. Not enough moisture and your cheese won’t melt properly in a sauce.
The technical name for the grains? Cheds. 
Flour may be your missing piece. I misremember where I learned this (may have been a good eats ep though) but the flour molecules will slip between the cheese protein strands and make a smoother sauce. Not sure if that is what is causing your grittiness, but worth a shot.
If you boil the sauce, it will be become grainy. You can fix it by putting an ice cube in.
You didn’t mention that in your OP. Someone is likely to correct me, but it seems a little TOO basic: how on earth do you make a bechamel sauce without starting with a roux? You need the flour to absorb the fat and hook up with the proteins into itty bitty grilled cheese sandwiches. Didn’t your mother teach you ANYTHING? Okay, mine didn’t teach me that one, either. Good cook, when she had Velveeta, but otherwise her cheese sauces separated into oil and grits.
Even I know how to avoid that, and tonight I served my family a fake Chinese Restaurant dinner consisting of Banquet chicken nuggets (despite my Minnesota roots, I refrained from serving the nuggets with cheese) with a nice sauce from Chun King (a fine brand from Duluth, Minnesota Prefecture), fried rice (one part Beef Rice-a-Roni for that hydrolyzed vegetable protein bang, two parts regular rice, and half a bag of frozen peas and carrots (for that genuine-fake Chinese buffet touch), and “Chinese” frozen Italian green beans with butter and garlic Teriyaki sauce.
Nobody complained, but that they had just returned from a day at the aquarium and didn’t have to cook in 87-degree weather may have limited their pickiness.
I get the sense that the OP is using a recipe that doesn’t make use of a bechamel sauce.
Michaela makes a “sort of” mac and cheese this way. She cooks the macaroni, then tears up twelve American cheese sandwich slices into it (after draining, of course), and stirs them into the pasta. She adds a little milk, and stirs it all together over a low heat, until the cheese is all melted.
I have no idea how one would convert this technique into the preparation of a stand-alone cheese sauce, though.
The question is, to squish or not to squish the itty bitty grilled cheese sandwiches?
Ouch, stop hitting me.
I have to say, my mother never made a homemade cheese sauce, but this thread is inspiring me to try. :o ← orange color indicates cheesiness, mouth is open the better to eat, one of my favorite activities