I feel you have fuel restriction
if you choke or prime the engine and it starts but will not continue to run w/o the choke on. Give more history:

1. Is this a sudden problem?

2. Are you using treated fuel
to help prevent ethanol fuel problems?

3. When was the last good complete
engine tune-up & maintenance (replaced plug, air cleaner, inspected all fuel
lines, cleaned and adjusted carb)?

Today’s engines seem to need
less maintenance but when mine fail to start easy I know instantly something
has changed or is needed. Starting with the
easy known needs (fresh fuel if it was not treated, Ck for a good clean spark, Ck
for good fuel lines/cln filter, Ck for a clean air filter). Then get into more detail Cks if needed
(Carb, Compression and Valve Adjustments).

This machine was made in 1996. I bought it about 10 years ago. The 1st owner was an older gentleman I believe he did not use it extensively. It was hard to start when I bought it.

I used it regularly the first year. Then only a couple of times each year. It was stored in outdoor shed. Starting has been always a problem. Regular fuel, never any additives.

There has not been any real tune-up. It ran when I could start. Eventually the engine would start and kept running.

But this time, it is harder to keep the engine running without stopping. Fuel should be ok, as well the air filter and spark plug. Fuel line can be an issue. I do not know if there is any good fix, or I should replace the entire fuel line(s). I think I should check and clean the carb before I replace anything else.

The carb is a good place to
start. Mainly, if you are not using
Stabil or Etc additives in your fuel. If
it’s Ethanol type fuel and sits for periods with un-treated fuel. I would look close for 2 things +, deteriorated
fuel line material and a swollen needle seat if they are rubber. Ck for a fuel filter that could be restricting
fuel vs screens in two cycles. A swollen
rubber needle’s tip or seat might not be noticeable to the eye. My experience was slightly noticeable but it
would stop the mower as if it was out of fuel with plenty of fuel in the tank. Please post Pics of what you find.

I would use compressed Carb
Cleaner with a stem and fine wire to verify all passages are clean. “Note” a
ref of all carb adjustments by counting turns to close lightly to know to come
back to those as ref starting points with new adjustments. I feel and hope the needle adjustment is good,
but swollen rubber on needle or in seat could be an issue.

Good Luck!

Since I am in the process of replacing a carburetor, I may not be the best person to give advise, but I think I would start with loger's second post rather than the far more difficult first post. For one thing, main jets seem to get a little perhaps unseen crud in them. I suppose your carb has one of those bowls with the drain nut in the bottom. While I would take the carb off and blow and clean out everything I could get to, you might just start by removing that nut and rinse out the bowl and run a wire up the bottom of the central pipe that will clear the main jet. The repairman 'helping' me with my carb specifically suggested the above.

BTW, those little rubber seats for the float valve need removal with something like a small drill bit to act as a pick, not a drill. Sometimes, someone with poor eyesight might end up with two seats on top of each other!

I believe in order to clean the carb, I'll need to remove it first. Since it costs about $20 to $30 to get a new carb, I may just get a new one instead. I'm not sure if I can remove the screw at the bottom of the carb. I think there is a spring near the screw....

I doubt your carb cost is correct, but if true then go for it; and then you can play with the other one. However, it is usually easy to drop the carb bowl or even leave/hold the bowl on and do the wire from the bottom cleaning that I mentioned above. It is usually a very easy first check without even removing the carb.

More points related I got
from the Tech about swollen seats:

1. He tries to use compressed
air to blow them out to be reused because any tear ruins it.

2. Tecumseh carbs with bowls
have various jets and bolts holding the bowl ported. It’s good to have a diagram to know what
should be cleaned or replaced where.

3. I hate spending a dime
unless I know it’s the needed Fix! Too Much
Stuff Now, LOL!

This very likely has a removable emulsion tube in the stem which seals must be good as well. The cheapo carb price is typically aftermarket.

Looking at some possible Carb
rebuild kits at the address below, you can see there is a little involved in
these type carbs. Most of all you will
see there is a rubber seat that could swell, passages and jets that could be
restricted and extra parts for different models.

Since my start in 1970 I have
not felt good about removing the round metal Welch Plugs. I have
been lucky enough not to have to remove one to solve a problem. The few small plugs appear related to porting
vs any mechanics. If I see good in/out compressed carb cleaner I feel it's
clear. Do you have a diagram of your
carb and does it show what is under the welch if a welch is present. Please post a pic or link to your carb’s
diagram.

http://www.amazon.com/s/?ie=UTF8&keywords=tecumseh+10+hp+carburetor&tag=googhydr-20&index=aps&hvadid=68291549489&hvpos=1s2&hvexid=&hvnetw=g&hvrand=17924540655184241384&hvpone=&hvptwo=&hvqmt=b&hvdev=c&ref=pd_sl_7h1wqosjtp_b