Slitting saws for steel are usually made of highspeed steel. Sounds like you are either spinning the saw too fast (60 rpms ??), cutting too deep (no more than the blade thickness deep), not using flood coolant, pushing the cut too fast, or a mix of the above. Also possible is that the metal you are cutting is a tough alloy or has some residual hardness (file test ??).
The flood coolant helps clear the fine saw teeth of chips as well as cool things down.
I have burned up a number of thin saw cutters spinning them way too fast. Check for the proper surface speed of the teeth. Using a slitting saw in a mill is a "form cut" not single point. Form cuts require much greater attention per above as they cause a higher chip load.
Last resort... yes, you could use fiber abrasive discs in your mill (at higher RPMs) but with shallow cuts and/or slow feed. Those fiber abrasive discs are available in many handy sizes. Even Harbor Freight has them. Keep the sliding mill surfaces protected and cleaned of grit/dust.
Do check the hardness of your slitting saw discs with a file. The file should slip over them without scratching the blades. A Dremel with abrasive discs and some patience you can re-sharpen your slitting saw blades. The small purchase price you mention sounds terriffic.