Does it matter if my underground conduit overlaps or crosses over existing conduits?
Crossing existing utilities is no problem; it happens all the time. Just be careful when digging. There is probably a prescribed margin to either side of the marks within which digging must be done with &quot;hand tools only.&quot; In some areas the margin is 24 inches to either side (48 inches plus the width of the marks themselves); in other areas it is 18 inches to either side; yet others may choose some other distance.
Even when hand digging a shovel can easily damage the insulation or even cut through direct-buried cable (telephone and CATV are often direct-bury; power might or might not be). Plastic pipes, such as gas pipe or electrical conduit, are much more resilient to impact from a hand-operated shovel.
Soils vary quite a lot. If yours is sandy then hand-digging will be easy. You could even vacuum excavate with the help of a wet-dry &quot;shop vac.&quot; Vacuum excavation is considered &quot;hand tools.&quot; If your soil is harder, like clay, it's going to be tough going. In this case I'd suggest digging the trench full-depth on both sides of the marks first. You could then make a choice: chip away at the remaining wall of hard soil, or simply bore through it at the height where the new conduit should go. This can be done with running water from a garden hose if the soil isn't too hard, or with a boost from a pressure washer. &quot;Hydro excavation&quot; as this is called is also counted as  &quot;hand tools&quot; and is used extensively by professional excavators. Its likelihood of damaging existing buried lines is very low.
There are tricks one can use to improve the DIY hydro excavation.

Use a pressure washer turbo nozzle. It shoots a 0 degree stream but orbits in a circle, carving a cone into the soil. It works much better than a plain 0 degree or fan-type nozzle.
Push a 2&quot;-2.5&quot; sleeve (pipe) into the bore as you go. It &quot;saves your progress&quot; each time you push it in deeper, defending against cave-in. It also eliminates concern of excess erosion of the bore because water and spoils exit through the sleeve. The sleeve must be large enough that spoils, including small stones, can flow past the boring device (pressure washer wand or hose/pipe) to exit the sleeve. Vacuuming with a wet/dry vac can help.
Bore at an uphill angle so that gravity will carry water and spoils back to you.
A pressure washer wand can easily be extended. Big box stores have begun carrying adapters from pressure washer quick-connect to regular pipe thread; a pair of these and one or more arbitrary-length 1/2&quot; threaded pipes joined with couplers works well. Ready-made extension parts are sold by outfitters for the utility hydro-excavation industry.

I've used these techniques to bore about 25 feet under my own driveway. The bore began in clay, but my upward angle was too steep and I ran into the gravel beneath the concrete and then into the concrete itself. With some creativity and persistence I got it to follow the bottom side of the concrete and exit at the far side.
Now I own some real boring equipment and do it on the side semi-professionally. :-)