Why did you go to HUD? Why not the Hartford Housing Authority?
http://www.hartfordhousing.org/
Don't pin your hopes on getting a book published - although I wish you well if you do. The market has been flooded with memoirs ever since the (excellent) Angela's Ashes. But who knows? You could luck out. Try starting locally and approaching the Courant with a column proposal. It might turn into a series if they're interested. Or any of the number of reputable cancer journals - even a Facebook group could steer you in the right direction.
It disturbs me what HUD told you. More typically if someone is sick (even if in remission) emergency public housing can be found, whether in a "project" or a private Section 8 apartment. I recently knew of a man who was living under a bridge, had gall bladder surgery, while recovering was placed in senior housing - usually an 8-year waiting list here in Boston.
Talk to the local housing authority, not the feds. See what they say.
Also, you say you will have $3,500 for this hike. If you apply for housing before the hike, you will have to spend that down. You are not allowed to have more than $2,000 from any and all sources to qualify for housing.
You need an advocate by your side. Many non-profits that have a religion in their name actually cater to everyone. Make some calls to Catholic Charities and Jewish Family & Childrens' Service. Drop in on a priest. Go to the Hartford-area shelters and speak to a social worker. People get housing more quickly when they have the support of a social worker; therapist; clergyperson.
People have posted to gofundme for less dire circumstances. But make sure you provide an avenue so they can check out your story.
I do know someone financing her hike by contracting with a non-profit PR group to speak. Lots of planning needed for that; she didn't find someone overnight; she networked and networked.
Seminar ... read a book. Well, Cheryl Strayed read a book and then did the PCT. It happens. I did the Long Trail without much hiking experience (but had done the Bikecentennial before then, and long-weekend hikes in New England). IDK. It's been done.
As for writing - you need to learn to write, first. And that includes spelling. I noticed a few spelling errors in your post. Yes, I know this is a blog and people often write quickly without proofreading first when they post to blogs - I know I do. But to be taken seriously as a speaker; memoirist; columnist - you will have to have perfect spelling, near-perfect grammar (colloquialisms are OK). Why not start here practicing your spelling? (Spell-check doesn't count, and it doesn't take homophones into account.)
illabelle and Traveler - I feel he needs SSDI, at least temporarily (SSDI has a "Ticket to Work" program which helps reintroduce people to the workforce. It sounds like the OP needs a lot more information than s/he's been given, and that's where a social worker comes into play. The oncologist can make a referral.